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TOP SCORERS FOR MIDTERM ( February 2020 )

TOP SCORERS FOR MIDTERM

Given on February 19 – 21,  2020

     MW 101 – Math in the Modern World (200 students)

                    Name                              Score

     1.  Reighxyne Joules Viado           100

          Jose Jerry Labrador                  100

          Queenie Rose Orong                 100

          Johana Dalamban                      100

          Nasria Talusob                           100

     2.  Maria Nina Nicole Singco           98

     3.  Saharia Kusain                            96

     4.  Ronnilyn Yambao                        90

     5.  Novelyn Balbuena                       88

     6.  Trisha Ramirez                             87

     7.  Monalisa Talusob                        86

     8.  Sittie Aisa Magao                        85

          Rohaina Laguiab                         85

          Christine Jane Ursua                  85
          Justine Ericka Bayog                  85
          Chennie Marianne Galleto          85
          Daryl Hannah Buagas                 85
          Baby Jane Dumamba                  85
          Josel Esparagoza                        85
          Melody Cadangin                         85 
     9.  April Rose Del Rio                       84
          Angel Valerie Ledda                    84
          Jame Choella Pitulan                  84
   10.  Joezeil Pagulong                         81
          Jenelyn Melocotones                  81            
           Joeseph Caalim                          81
           Kenneth Andea                           81
          Reymark Calambro                      81
          Nicko Jay Andan                         81
          Honeylyn Zerrudo                       81
   11.  Danica Mae Singco                     80           

       MATH 106 – Logic and Set Theory ( 24 students )

                    Name                              Score

     1.  Sabila Mamalampay                    98

     2.  Bailyn Tukuran                             96

     3.  Danilo Derio Jr.                            92
     4.  Marvin Chiva                                89
     5.  Annie Rose Rosa                         88
     6.  Joseph Gabe                                87
     7.  Kenneth John Pernal                  85 
     8.  Ahmad Ulanan                             83



          
       

TOP SCORERS FOR TRINALS

TOP SCORERS FOR TRINALS

Given on December  17 – 18,  2019

       MW 101 – Math in the Modern World

      Name                                     Score
     1.  Trisha V. Ramirez              86
          Ronnilyn C. Yambao          86
          Mac Royce D. Diano          86
     2.  Reighxyne Joules Viado    84
          Rohaina  M.  Laguiab        84
     3.  Roosevelt  Ger Dojinog     82
     4.  Justine Ericka Bayog        81
     5.  Jose Jerry Labrador          80

  MATH  106 – Mathematical Logic and Set Theory

      Name                                     Score
     1.  Joseph B. Gabe                 100
        Danilo E. Derio Jr.            100
  Benladin S. Samier                 100
     2.  Dominic J. Martinez          99
     3.  Sabila M. Mamalampay     98
     4.  Datu Ali A. Pandaupan      97
          Annie Rose D.  Rosa         97
     5.  Jean P. Bautista                95
        Ahmad G. Ulanan             95
  Stephiene C.  Pasagui              95
  Kenneth John C. Pernal            95
     6.  Bhlyzyr Thryz  R.  Diaz    94
          Joshua G.  Larazan           94
  Bailanie  A.  Sumlay               94
     7.  Kaye J.  Bolina                91
     8.  Bailyn E.  Tukuran           89
     9.  Jonna Fe  J.  Gorit            87
   10.  Vaneza Joy Y. Largo         86
   11.  John Lloyd  C. Salvallon   85
   12.  Marvin C.  Chiva             84
   13.  Shery Mae  D.  Angcon     82


                                        Efren F. Cadungog, Sr. , MIM
                                                Class Adviser

MW 101 LESSON 2

SPEAKING MATHEMATICALLY
     2.1 Variables
             Variable is sometimes thought of as a mathematical “John Doe” because you can use it as a placeholder when you want to talk about something; either
             a) you imagine that it has one or more values but you don’t know what they are,
       b) you want whatever you say about it to be equally true for all elements in a given set and you don’t want to be restricted to considering only a particular, concrete value  for it.

 Example of the first use of a variable:
  Is there a number with the following property: doubling it and adding 3 gives the same result as squaring it?
            In this statement, we introduce a variable to replace the potentially ambiguous word “it”

                             Is there a number x with the property that 2 x  +  3 =  x²  ?

The advantage of using a variable is that it allows you to give a temporary name to what you are seeking so that you can perform concrete computations with it to help discover its possible values. To emphasize the role of the variable as a placeholder, you might write the following:

                             Is there a number    with the property that 2 × € + 3 = € ² ?

 The emptiness of the box can help you imagine filling it in with a variety of different values, some of which might make the two sides equal and others of which might not.

      To illustrate the second use of variables, consider the statement:
              No matter what number might be chosen, if it is greater than 2,
              then its square is greater than 4. 

       In this case introducing a variable to give a temporary name to the (arbitrary) number you might choose enables you to maintain the generality of the statement, and replacing all instances of the word “it” by the name of the variable ensures that possible ambiguity is avoided: 
       No matter what number n might be chosen, if  n  is greater than 2, then  n²  is greater than  4.

Example : Writing sentences using variables.
   a. Are there numbers with the property that the sum of their squares equals the square of their sum.
   b. Given any real number, its square is nonnegative.

Solution :
       a) Are there numbers  a and b with the property that  a² + b²  =  ( a + b)² ? 
              Or,  Are there numbers  a  and  b  such that  a² + b²  =  ( a + b)² ? 
              Or,  Do there exist any numbers  a  and  b  such that  a² + b²  =  ( a + b)² ? 

       b)  Given any real number  r,  r² is nonnegative.
              Or, For any real number  r,  r²  ³ 0.
        Or, For all real numbers r,  r²  ³ 0.

  Kinds of Mathematical Statements
        1. Universal statements
        2. Conditional statements
  3. Existential statements

    * A universal statement states that a certain property is true for all elements in a set.
          Ex 1.  All even numbers are multiples of 2.
          Ex 2.  All integers ending in 0 and 5 are divisible by 5. 

    * A conditional statement states that if one thing is true then some other thing also has to be true.   
            Ex 1.  If 1296 is divisible by 12, then 1296 is divisible by 6.
            Ex 2.  If I am thirsty, then I will drink.

    * An existential statement states that there is at least one thing for which the property is true.
             Ex 1. There is an x  in  x² – x – 6 = 0, such that a) x is odd,  b)  x is even.
             Ex 2. There is a prime number that is even. 


UNIVERSAL CONDITIONAL STATEMENTS
          Universal statements contain some variation of the words, “For all”, and conditional statements contain versions of the words, “if-then”. A universal conditional statement is a statement that is both universal and conditional.
     Ex. 1.  For all animals A, if A is a dog, then A is a mammal.

          A universal conditional statement can be written in ways that make them appear to be purely universal or purely conditional. The previous example can be transformed in a way that makes its conditional nature explicit but its universal nature implicit:
                      If A is a dog, then A is a mammal.
                    Or: If an animal is a dog, then the animal is a mammal.

         The statement can also be expressed so as to make its universal nature explicit and its conditional nature implicit:
                For all dogs A, A is a mammal.
    Or :  All dogs are mammals.

 Rewriting a Universal Conditional Statement

   Exercise 1. Fill in the blanks to rewrite the following statement.
    For all real numbers x, if x is nonzero then x²  is positive.
         a)  If a real number is nonzero, then its square _____.
         b)  For all nonzero real number  x, _____.
         c)  If x is  ______, then ______.
         d)  The square of any nonzero real number is ______.
         e)  All nonzero real numbers have ______. 
         
    Answers:
         a)  is positive.
         b)  x² is positive.
         c)  is a nonzero real number,   x²  is positive.
         d)  positive.
         e)  positive squares( or squares that are positive).


  UNIVERSAL EXISTENTIAL STATEMENTS
          A universal existential statement is a statement that is universal because its first part says that a certain property is true for all objects of a given type, and it is existential because its second part asserts the existence of something.
    Ex.  Every real number has an additive inverse.

      In this statement the property “has an additive inverse” applies universally to all real numbers. “Has an additive inverse” asserts the existence of something – an additive inverse – for each real numbers. The nature of the additive inverse depends on the real number, and the statement can be rewritten in several ways; some less formal and some more formal.
                    All real numbers have additive inverses.
      Or:  For all real numbers r, there is an additive inverse for r.
      Or:  For all real numbers r, there is a real number s such that s is an additive inverse for r.

Introducing names for the variables simplifies references further in discussion. For instance, after the third version of the statement, you may write :
 When r is positive, s is negative, when r in negative, s is positive and when r is zero, s is also zero.

One of the most important reasons for using variables in mathematics is that it gives you the ability to refer to quantities unambiguously throughout a lengthy mathematical argument, while not restricting you to consider only specific values for them.

REWRITING UNIVERSAL EXISTENTIAL STATEMENT
  Fill in the blanks to rewrite the following statement:
         Every pot has a lid.
        a)  All pots ______.                                                                    
        b)  For all pots P, there is a ______.                         
        c)  For all pots P, there is a lid L such that _______.     

   Answers:
        a) have lids.
        b) a lid for P.
        c) L is a lid for P.
   

EXISTENTIAL UNIVERSAL STATEMENTS
      An  existential universal statement is a statement that is existential because its first part asserts that a certain object exists and is universal because its second part says that the object satisfies a certain property for all things of a certain kind.  For example:
       There is a positive integer that is less than or equal to every positive integer.

    This statement is true because the number one is positive integer, and it satisfies the property of being less than or equal to every positive integer. The statement can be rewritten in several ways, some less formal and some more formal:
      Some positive integer is less than or equal to every positive integer.
  Or: There is a positive integer m that is less than or equal to every positive integer.
  Or: There is a positive integer m such that every positive integer is greater than or equal to m.
 Or: There is a positive integer m with the property that for all positive integers n, m £ n.

REWRITING AN EXISTENTIAL UNIVERSAL STATEMENT
  Fill in the blanks to rewrite the following statement in three different ways:
      There is a person in my class who is at least as old as every person in my class. 
   a) Some _____ is at least as old as _____.              
   b) There is a person P in my class such that P is _____.       
   c) There is a person P in my class with the property that for every person P in my class, p is ____.                                 
 Answers:
    a)  person in my class,  every person in my class.
    b)  at least as old as every person in my class.
    c)  at least as old as q.

Some of the most important mathematical concepts, such as the definition of a limit of a sequence, can only be defined using phrases that are universal, existential and conditional, and they require the use of all three phrases “for all”, “there is”, and “if – then”.
     Example:
            If  a₁, a₂, a₃,  . . .  is a sequence of real numbers, saying that
                         the limits of  a_n  as  n  approaches infinity is L
           means that
                        for all positive real numbers  e, there is an integer N such that
                      for all integers n,  if  n  >  N  then  –e  <   a_n – L <  e .

  Exercise set 2.1 

Fill in the blanks to rewrite the given statements.

  1.  For all objects O, if O is a square then O has four sides.

           a.   All squares  _____________ .

           b.   Every square ____________ .

           c.  If an object is a square, then it  __________ .

           d.  If  O  _____ , then  O _________ .

           e.  For all squares O,  ____________ .

 2.  For all equations E, if E is quadratic then E has at most two real solutions.

           a.  All quadratic equations ____________ .

           b.  Every quadratic equation ___________ .

           c.  If an equation is quadratic, then it _________ .

           d.  If E ________ ,  then E _______________ .

           e.  For all quadratic equation E,  __________ .

 3.   Every positive number has a positive square root.

          a.  All positive numbers ______ .

          b.  For any positive number  N, there is _______ for N.

          c.  For all positive numbers  N, there is a positive number r such that ______ . 

 4.  Every nonzero real number has a reciprocal.

         a.  All nonzero real numbers ______ .

         b.  For all nonzero real numbers  r,  there is ______ for  r.

         c.  For all nonzero real numbers r, there is a real number s such that _____.

  

  5.  There is a real number whose product with every number leaves the number unchanged.

         a.  Some ______ has the property that its ______ .

         b.  There is a real number r such that the product of  r  ______.

         c.  There is a real number r with the property that for every real number s, ______ .      

 Fill in the blanks using  variable or variables to rewrite the given statements.

1.    Is there an integer tha has a remainder of 2 when it is divided by 5 and a remainder of 3 when it is divided by 6?

a)       Is there an integer  n  such that n has _______ ?

b)      Does there exist  ________  such that if n is divided by 5 the remainder is 2 and if  ______ ?

c)       Give at least one example of this integer.   

2.    Given any two real numbers, there is a real number in between.

a)       Given any two real numbers  x  and  y, there a real number  r  such that r  is ______ .

b)      For any two ______ ,  _______ such that  x <  r  <  y.

     

3.    Is there a real number whose square is  – 1?

a)       Is there a real  number x such that  _____ ?

b)      Does there exist  ______ such that x² =  – 1 ?

c)       Give at least one example of this number.

4.    Given any real number, there is a real number that is greater.

a)       Given any real number  r ,  there is  _____  s  such that  s  is ______ .  

b)      For any ______ , _______ such that  s  >  r.

5.     The cube root of any negative real number is negative.

a)       Given any negative real number  s , the cube root of ______ .

b)      For any real number  s , if  s  is _____ ,  then _______ .

c)       If a real number  s  ______,  then  ______ .   

                        SETS

 1.2 The Language of sets 
      Georg Cantor (1845 – 1918) introduced the word set as a formal mathematical term in 1879.
A SET is a well-defined collection of objects. The objects are the members of the elements of the set.

Examples of sets : Well-defined sets                                              Not well-defined set
       1.  Set of solution to  x² –  7x  +  6 = 0.                                    1.  Set of presidents
       2.  Set of vowels of the English alphabet.                                   2.  Set of  subjects
       3.  Set of positive integers greater than 1 but less than 8              3.  Set of municipalities
       4.  Set of cities of Region XII.

  Notation:
       Sets are denoted by capital letters, such as A, B, C, D, X, Y . . .  whereas the lower case letters  such as  a, b, c, x, y . . . is used to denote members or elements of the set. 
       If S is a set, the notation x  S, means that x is an element of S. The notation x  S, means that x is not an element of S.
       A set may be specified using a set-roster notation by writing all the elements between braces.
Examples :
        If N is a set of countries of the UN, then the Philippines is an element of N.
        If B is a set of positive even integers greater than 1 but less than 8, then 4 is an element of B.
        If C is a set of cities of Region XII, then Koronadal City is an element of C.
                          Is Africa an element of N?
                          Is 5 an element of B?
                          Is Malaybalay City an element of C?,

      A variation of the notation is used to describe a very large set, as when we write
                          {1,   2,   3,   4, . . . ,  100}
   to refer to a set of all integers from 1 to 100. A similar notation can also describe an infinite set, as when we write {1, 2, 3, . . . } to refer to a set of all integers.
      The symbol  . . .  is called an ellipsis and is read “and so forth.”

    An axiom of extension says that a set is completely determined by what its elements are – not the order in which they might be listed or the fact that some elements might be listed more than once.

 Example 1 : Using the set-roster notation
   a.  Let A = {1, 2, 3}, B = {3, 1, 2}, and C = {1, 1, 2, 3, 3, 3}. What are the elements of A, B, and C? How are A, B, and C related ?
   b.  Is { 0 } = 0?
   c.  How many elements are in the set {1, { 1 } } ?
   d.  For each nonnegative integer n, let U_n = { n,  – n}. Find U₁, U₂ and U₀.  

Solution :
    a.  A, B, and C have exactly the same three elements: 1, 2, and 3. Therefore, A, B, and C are simply different ways to represent the same set.
       
    b.  {0} ¹ 0 because {0} is a set with one element, namely 0, whereas 0 is just the symbol that
         represents the number zero.
    c.  The set {1, { 1 } } has two elements: 1 and the set whose only element is 1.
    d.  U₁ = { 1,  – 1},  U₂ = { 2, – 2}, U₀ = { 0, – 0} = { 0, 0} = { 0 } 

Certain set of numbers are so frequently referred to that they are given special symbolic names. These are summarized in the table below.

The set of real numbers is pictured as the set of all points on a line as shown on the real number line. The number 0 corresponds to a middle point, called the origin. The set of real numbers is divided into three parts: the set of positive real numbers, the set of negative real numbers, and the number 0. Note that 0 is neither positive nor negative.

The real number line is continuous because it is imagined to have no holes. Every integer is a real number, and because the integers are all separated from each other, the set of integers is discrete.


SET–BUILDER NOTATION
Another way to specify a set uses a set-builder notation.   Let S denote a set and let P(x) be a property that elements of S may or may not satisfy. We may define a new set to be the set of elements x in S such that P(x) is true. The set is denoted by

Occasionally we write { x | P(x) } without being specific about where the element x comes from.

ORDERED PAIR

        Given elements a and b, the symbol (a, b) denotes the ordered pair consisting of  a and b together with the specification that  a  is the first element of the pair and  b  is the second element. Two ordered pairs (a, b) and (c, d) are equal if, and only if a = c and b = d.

       Symbolically:

                            (a, b) = (c, d) means that a = c and b = d.

CARTESIAN PRODUCT

       Given sets  A and B,  the Cartesian product of A and B, denoted by  A x B  and read as “A cross B”, is the set of all ordered pairs (a, b), where a is in A and b is in B. Symbolically:

                    A x B  = { (a, b) | a Î A and b Î B }

  Example 6: Cartesian product

            Let  A = { 1, 2, 3 }  and  B = { m, p }

     a.  Find A x B

     b.  Find B x A     

     c.  B x B

     d.  How many elements are in  A x B,  B x A  and  B x B?

     e.  Let R denote the set of all real numbers. 

          Describe R x R.

Solution :

  a.  A x B =  { (1, m), (2, m), (3, m), (1, p), (2, p), (3, p) }

  b.  B x A =  { (m, 1), (m, 2), (m, 3), (p, 1), (p, 2), (p, 3) }

  c.  B x B =  { (m, m), (m, p), (p, m), (p, p) }

  d. A x B has six elements. This is the number of elements in A times the number of elements in B.

B x A has six elements, the number of elements in B times the number of elements in A.

B x B has four elements, the number of elements in B times the number of elements in B.

 e. R x R is the set of all ordered pairs (x, y) where both x and y are real numbers. If horizontal and vertical axis are drawn on a plane and a unit length is marked off, then each ordered pair in R x R corresponds to a unique point in the plane, with the first and the second elements of the pair indicating the abscissa and ordinate the point.

       The term Cartesian plane is often used to refer to a plane with this coordinate system, as shown in the figure:  

  1.3 The Language of Relation and Function    

            There are many kinds of relationships in the world. Two people are related by blood if they share a common ancestor, we also have a relationship between student and teacher,  between people who work for the same employer, and between people who have a common ethnic background.

            The objects of mathematics maybe related in many ways. A set B maybe related to a set C if B is a subset of C, or B is not a subset of C, or B and C have a common element. A number x is related to a number y if  x < y, or if x is a factor of y, or if  x² + y² = 1.

            Identifiers in a computer program are related if they have the same first eight characters, or  the same memory location is used to store their values when the program executes.

            Let A = {0, 1, 2} and B = {1, 2, 3 }. An element x in A is related to an element y in B if, and only if, x is less than y.  The notation  xRy is shorthand for the sentence “x is related to y”. From the given set:

                          0 R 1   since  0 <  1

                          0 R 2   since  0 <  2

                          0 R 3   since  0 <  3

                          1 R 2   since  1 <  2

                          1 R 3   since  1 <  3   and

                          2 R 3   since  2 <  3

Lesson 1 MATH 106 - Logic and Set Theory

LOGIC

It is not easy to summarize in a few paragraphs the subject matter known as logic. For lawyers and judges, logic is the science of correct reasoning. They use logic to communicate more effectively, construct valid arguments, analyze legal contracts, and make decisions. Law schools consider knowledge of logic to be one of the most important predictors of future success of their new students. A sizable portion of the LSAT (Law School Admission Test) considers logical reasoning as part of their admission process.    

Other professions also make extensive use of logic. Programmers use logic to design computer software, electrical engineers use logic to design circuits for smart phones, and mathematicians use logic to solve problems and construct mathematical proofs. In this chapter, you will encounter several facets of logic. Specifically, logic is use to

·   Analyze information and the relationship between statements

·   Determine the validity of arguments

·   Determine valid conclusions based on given assumptions, and

·   Analyze electronic circuits.

LOGIC STATEMENTS AND QUANTIFIERS

         Gottfried Wilhelm Leibniz (1646 – 1716) was one of the mathematicians who make a serious study of symbolic logic. He tried to advance the study of logic from a merely philosophical subject to a formal mathematical subject. Leibniz  never completely achieve his goal; however several mathematicians such as Augustus De Morgan (1806 – 1871) and George Boole (1815 – 1864), contributed to the advancement of symbolic logic as a mathematical discipline.

         Boole published “The Mathematical Analysis of Logic” in 1848. In 1854 he published the more extensive work, “An Investigation of the Laws of Thought”. With these documents, the mathematician Bertrand Russell stated, “Pure mathematics was discovered by Boole in a work which is called The Laws of Thought.”  

LOGIC STATEMENT

     Statement or Proposition is a declarative sentence that is either true or false, but not both true and false. It is typically expressed as a declarative sentence (as opposed to question or command). Propositions are the basic building blocks of any theory of logic.

 Every language contains different types of sentences, questions, and commands. For instance,

            “Is there a rain today?”  is a question.

            “Go and set that bird free” is a command.

            “This is a nice design” is an opinion.

            “Pasig City is the capital of the province of Rizal” is a statement of fact.

The symbolic logic that Boole was instrumental in creating applies only to declarative sentences.

Examples.

   Determine whether each sentence is  statement.

      1.  Mexico is in Africa.

      2.  How are your parents ?

      3.  9⁹ + 2  is a prime number.

      4.  x + 2  =  4

Answers:

      1.  Statement with a truth value of false because Mexico in Central America.

      2.  It’s a question and not a declarative sentence, hence not  statement.  

      3.  You may not know that 9⁹ + 2  is a prime number, however, you know that it is a whole number larger than 1, so it is either a prime number or it is not a prime number. The sentence is either true or false, and it is not both true and false, so it is a statement.

      4.  x + 2  =  4  is a statement, known as an open statement and is true only for x = 2, and false for any other values of x.

Exercises:

Which of the following are statements or proposition ?

     1.  The only positive integer that divide 5 are 1 and 5 itself.

     2.  2  +  4 = 7

     3.  The earth is an oblate spheroid.

     4.  4 – x = 7.

     5.  Do you speak Chinese? 

     6.  Oh, what a beautiful site!

     7.  Buy three tickets for the concert on November 25, 2018.

     8.  The only positive integer that divide 12 are 3, 4 and itself.

     9.  A square is a rectangle having all sides equal.

   10.  A circle is round.

SIMPLE STATEMENTS AND COMPOUND STATEMENTS

       A simple statement is a statement that conveys a single idea. A compound statement is  statement that conveys two or more ideas. Connecting simple statements with words and phrases such as and, or,  not, if...then, and if and only if creates a compound statement.  For instance, “I will go to school and I will join the debate” is a compound. It is composed of two simple statements “I will go to school” and “I will join the debate.” The word and is the connective for the two simple statements.

George Boole used the letters p, q, r, and s  to represent simple statements and the symbols Ù,  Ú,  ~,  ®  and  «  to represent connectives.

Let us define the meaning of these connectives by showing the relationship between the truth value (i.e. true or false) of composite propositions and those of their component propositions.

Let p and q be propositions. 

   a) The conjunction of p and q, denoted by p L q, is the proposition  p and q.

   b) The disjunction of p and q, denoted by p V q, is the proposition p or q.  

   c) The negation of p, denoted by ~p, is the proposition not p.

        Propositions such as p L q and p V q that result from combining propositions are called compound propositions. The compound statement p L q is true when both p and q are true; otherwise, it is false. The compound statement  p V q is true if at least one of p or q is true, it is false when both p and q are false.

       The truth value of a simple statement is either true ( T) or false ( F). The truth value of a compound statement depends on the truth values of its simple statements and its connectives.  The truth values of propositions such as conjunctions and disjunctions can be described by truth tables. The truth table shows the truth value of compound statement for all possible truth values of its simple statements.

      The truth table of a proposition p made up of the individual propositions p₁...p_n, lists all possible combinations of truth values for p₁...p_n ,T denoting true and F denoting false and for each such combination lists the truth value of p.  

NEGATION

Write the negation of each statement.

    1.  p: Today is not Friday.   

       ~p:  Today is Friday.

    2.  p: The seventh month is July.   

       ~p: The seventh month is not July.

   3.   p:  Mathematical logic is not an easy subject.   

   4.   p:  Tacurong City is a city in the province of Sultan Kudarat.   

   5.   p:  The national flower is sampaguita.   

   6.   p:  Tarsiers are abundant in the province of Bohol.   

   7.   p:  Vatican City is a city within a City.   

   8.   p:  Japan is the land of the rising sun.   

   9.   p:  Cagayan de Oro City is known as the City of Golden Friendship.   

10.   p:  Cebu City is the Queen City of the South.   

CONJUNCTION, Ù   and DISJUNCTION, Ú

 Given  p:  Today is Monday.   

            q:  The weather is cold.

             r:  I passed the test.

             s:  I will be very happy.     

  1.  p Ù q :  Today is Monday  and the weather is cold.

  2.  r Ù s :  I passed the test  and I will be very happy.

  3.  p Ú q :  Today is Monday or the weather is cold.

  4.  r Ú s :   

  5.  ~p Ù q :

  6.  r Ù ~s :    

  7.  p Ú ~q :

  8.  ~r Ú s :  

  9.   q Ú s  

10.   p Ù r

11.   q Ú ~s  

12.   ~p Ù r 

Consider the following simple propositions.

    p: Today is Tuesday.

    q:  It is a sunny day.

    r:   I am going to the gym.   

    s:  I am not going to play basketball.

Write the following compound statements in symbolic form.

   1.  Today is Tuesday and it is a sunny day.

   2.  It is not a sunny day and I am going to the gym.

   3.  I am going to play basketball or I am not going to the gym.

   4.  It is not a sunny day and today is not Tuesday.

   5.  I am not going to play basketball or I am not going to the gym.

   6.  I am going to play basketball and I am going to the gym. 

   7.  It is not a sunny day and today is Tuesday.

   8.  I am going to play basketball and today is Tuesday.

Answers :

    1.  p Ù q

    2.  ~q Ù r 

Consider the following statements

      p:  The game will be played in New York.

      q:  The game will be shown on ABC 5.

      r:   The game will not be shown on ABS-CBN.

      s:   The GSW  are favored to win.

Write each of the following symbolic statements in words.

   1.  p Ù q                                6.  ~p Ù ~q

   2.  q Ù p                                7.  q Ù ~p

   3.  p Ú ~r                              8.  ~p Ú ~r  

   4.  q Ù s                                9.  s Ù q

   5.  q Ú ~ r                           10.  ~q Ú  r