MATHEMATICS LOGIC AND SET THEORY PRESENTATION.pptx

Gottfried Wilhelm Leibniz tries to advance
the study of logic from a mere philosophical
subject to a formal mathematical subject.
Leibniz never completely achieved this goal;
however, several mathematicians, such as
Augustus de Morgan and George Boole,
contributed to the advancement of symbolic
logic as a mathematical discipline.

LOGIC
¾Logic is the study of the
methods and principles of
reasoning.

LOGIC
STATEMENT
¾ A logic statement or proposition is a
declarative sentence that is true or false
but not both.
¾ There must be no ambiguity.
¾ In logic, the truth of a statement is
established beyond ANY doubt by a well-
reasoned argument.

LOGIC
STATEMENT
Examples:
a. You will pass the licensure
examination for teachers.
b. You will be a topnotcher.
c. You did well today.
d. It’s okay not to be okay.
e. Loving him was red.

LOGIC
STATEMENT
Exercise: Determine whether each sentence
is a statement.
1.Do you think you'll pass the LEPT?
2. I love Philippines.
3. Wena is a good dancer.
4.Did he cheat on Kath?
5.Please give me another chance.
6.

SIMPLE STATEMENT
 A simple statement is a statement that
conveys a single idea.
Examples:
a) Zero times any real number is zero.
b) 1+1=2.
c) All birds can fly

COMPOUND STATEMENT
 A compound statement is a statement
that conveys two or more ideas. It
contains several simple statements. The
ideas in a compound statement are
connected by connectives.

 Mathematical statements may be joined by
logical connectives, such as and, or, if . . .
then, and if and only if, which are used to
combine simple propositions to form
compound statements.
 These connectives are negation,
conjunction, disjunction, implication, and
biconditional.
LOGICAL
CONNECTIVES

Examples:
a)The grass is green and the sky is blue.
b)It is cold or it is sunny.
c)If a person is kind, then he is helpful.
d)The number 12 is an even number if
and only if it is divisible by 2.
LOGICAL
CONNECTIVES

Statements can be represented by propositional variables , .
𝒑 𝒒
LOGICAL
CONNECTIVES
LOGIC SYMBOLS NOTATION MEANING
Negation
Conjunction
Disjunction
Conditional /
Implication
Biconditional

The negation of a statement is the opposite of a given
mathematical statement.
Examples:
a)2 is the smallest prime number.
b)I am feeling well tonight.
c)I am cute.
NEGATION OF
STATEMENT
2 is not the smallest prime number.
I am not feeling well tonight.
I am not cute.

WRITING COMPOUND
STATEMENTS IN
SYMBOLIC FORM
Consider the following simple statements.
p: Today is Tuesday.
q: It is raining.
r: I am going to a movie date.
s: I am not going to a basketball game.
Write the following compound statements in symbolic form.
Today is Tuesday and
it is raining. 𝑝 ⋀ 𝑞

WRITING COMPOUND
STATEMENTS IN
SYMBOLIC FORM
q: It is raining.
It is raining and I am
going to a movie date. q ⋀ 𝑟

WRITING COMPOUND
STATEMENTS IN
SYMBOLIC FORM
q: It is raining.
I am going to the basketball game
or I am going to a movie date.
∼ 𝑠 ∨𝑟

WRITING COMPOUND
STATEMENTS IN
SYMBOLIC FORM
q: It is raining.
If it is raining, then I am not
going to the basketball game.
𝑞 → 𝑠

TRANSLATE SYMBOLIC
STATEMENTS
p: The pageant will be held in Manila.
q: The pageant will be televised on ABS-CBN.
r: The pageant will not be shown in GMA.
s: The Philippines' candidate is favored to win.
The pageant will be televised on
ABS-CBN and it will be held in
Manila.
𝑞 ∧ 𝑝

TRANSLATE SYMBOLIC
STATEMENTS
The pageant will be shown in
GMA and the Philippines'
candidate is favored to win.
𝑟 ∧ 𝑠

TRANSLATE SYMBOLIC
STATEMENTS
If the Philippines' candidate is
favored to win, then the pageant
will be held in Manila. .
𝑠 → 𝑝

A conditional statement consists of two
parts, a hypothesis in the “if” clause and a
conclusion in the “then” clause.
Every conditional statement has three
related statements. For every implication or
conditional statement ( ), we can construct
𝑝→𝑞
its converse, inverse, and contrapositive.
CONVERSE, INVERSE,
& CONTRAPOSITIVE

CONVER
To form the converse of the
conditional statement ( ),
𝑝→𝑞
interchange the hypothesis and
the conclusion. ( )
𝑞→ 𝑝

INVERSE
To form the inverse of the
conditional statement ( ), take
𝑝→𝑞
the negation of both the hypothesis
and the conclusion.

CONTRAPOS
To form the contrapositive of
the conditional statement ( ),
𝑝→𝑞
interchange the hypothesis and the
conclusion of the inverse statement.

Example: If I get a job, then I can help my
parents.
Converse: If I can help my parents, then I get a
job.
Inverse: If I don’t get a job, then I cannot help
my parents.
Contrapositive: If I can’t help my parents, then I
won’t get a job.
CONVERSE, INVERSE,
& CONTRAPOSITIVE

TRUTH TABLES,
TAUTOLOGIES,
LOGICAL
&

 Mathematicians normally use a two-valued
logic: Every statement is either True or False.
This is called the Law of the Excluded Middle.
 A statement in sentential logic is built from
simple statements using the logical
connectives , , , , and . The truth or falsity of a
statement built with these connectives
depends on the truth or falsity of its
components.

A truth table shows how the truth
or falsity of a compound statement
depends on the truth or falsity of
the simple statements from which
it's constructed.

TRUTH TABLE FOR
NEGATION
If P is true, its negation is false. If P is
false, then is true.
T F
F T

TRUTH TABLE FOR
CONJUNCTION
should be true when both P and Q are
true, and false otherwise.
T T T
T F F
F T F
F F F

TRUTH TABLE FOR
DISJUNCTION
is true if either P is true or Q is true or both.
It's only false if both P and Q are false.
T T T
T F T
F T T
F F F

TRUTH TABLE FOR
CONDITIONAL
The statement “if P then Q” is true if both P
and Q are true, or if P is false.
T T T
T F F
F T T
F F T

TRUTH TABLE FOR
BICONDITIONAL
means that P and Q are equivalent. So, the double
implication is true if P and Q are both true or if P and Q
are both false; otherwise, the double implication is false.
T T T
T F F
F T F
F F T

THINGS TO
REMEMBER
When constructing a truth table,
do consider all possible assignments of
True (T) and False (F) to the
component statements. Each of these
statements can be either true or false,
so there are possibilities.

THINGS TO
REMEMBER
To avoid duplication or omission
in assigning truth values to the
component statements, the easiest
and most systematic approach is to
use lexicographic ordering.

THINGS TO
REMEMBER
Example: For a compound statement with
three components P, Q, and R, here are the possible
assignments:

THINGS TO
REMEMBER
There are different ways of
setting up truth tables. For instance,
write the truth values "under" the
logical connectives of the compound
statement, gradually building up to the
column for the "primary" connective.

EXAMPLE
Construct a truth table for the compound
statement :
T T F T F
T F F F F
F T T T T
F F T T T

TAUTOLOGY
A tautology is a formula that is
"always true“, that is, it is true for
every assignment of truth values to
its simple components. Think of
tautology as a rule of logic.

CONTRADICTI
ON
A contradiction is false
for every assignment of
truth values to its simple
components.

EXAMPLE
Show that is a tautology.
T T T T T
T F F T T
F T T F T
F F T T T

LOGICALLY
EQUIVALENT
Two statements X and Y are
logically equivalent if is a tautology.
Another way to say this is: For each
assignment of truth values to the
simple statements that make up X
and Y, the statements X and Y have
identical truth values.

EXAMPLE
Show that and are logically equivalent.
T T T F T
T F F F F
F T T T T
F F T T T

TAUTOLOGIES &
LOGICAL EQUIVALENCES
When a tautology has the form of a biconditional, the two
statements that make up the biconditional are logically
equivalent. Hence, you can replace one side with the other
without changing the logical meaning.

SET
¾ A SET is a collection of well-defined objects.
¾ The objects in the set are called the
ELEMENTS of the set.
¾ To describe a set, we use braces { }, and use
capital letters to represent it.
¾ To indicate membership, we use the symbol
, when an element is not a membership,
∈
we use .

SET
Examples:
A = {2, 4, 6, 8, 10}
B = {all licensed professional teachers}
C = { }
D = {consonants of the English alphabet}
E = {Instagram, Facebook, X, TikTok}
F = {x ∈ N | x < 5}

SET
REPRESENTATIO
N
Recursive Rule
― By defining a set of rules which
generates or defines its members.
Examples:
D = {consonants of the English
alphabet}

SET
REPRESENTATIO
N
Listing / Roster Method
― Writing or listing down all the
elements between braces.
Examples:
A = {2, 4, 6, 8, 10}

SET
REPRESENTATIO
N
Set-Builder Notation
― Enumerating its elements by stating
the properties that its members must
satisfy.
Examples:
•C = { }
•F =

 A finite set contains elements that can be
counted and terminates at a certain natural
number.
Examples of Finite Set:
A = {2, 4, 6, 8, 10}
D = {consonants of the English alphabet}
F = {x N | x < 5}
∈
FINITE SET

 An infinite set is a set whose elements can
not be counted. An infinite set is one that has
no last element.
Examples of Infinite Set:
C = { }
INFINITE
SET

 This is a set with no elements, often
symbolized by or { }.
∅
Examples:
G = {vowel in the word “CRYPT”}
G = ∅
NULL SET

 A set with only one member.
Examples:
H = {number that is an even prime number}
H = {2}
SINGLETON
SET

 Two sets are equal if they contain the same
elements.
Examples:
F = {x N | x < 5}
∈
I= {1, 2, 3, 4}
EQUAL SETS

 Two sets are equivalent if they contain the
same number of elements.
Example:
F = {x N | x < 5}
∈
EQUIVALENT
SETS

 A set that contains all the elements
considered in a particular situation and
denoted by U.
Example:
U = {letters of the English alphabet}
J = { b , c , d , e , f , g , h , i , j }
UNIVERSAL
SET

 A set A is called a subset of B if every
element of A is also an element of B. “A
is a subset of B” is written as A  B.
 ∅ is a subset of every set.
 A set is always a subset of itself.
SUBSET

Example: I = { 1, 2, 3, 4}
Subsets:
{ 1 }, { 2 }, { 3 }, { 4 },
{ 1, 2 }, { 1, 3 }, { 1, 4 }, { 2, 3 }, { 2, 4 }, { 3, 4 },
{ 1 , 2 , 3 }, { 1 , 2 , 4 }, { 1 , 3 , 4 }, { 2 , 3 , 4 },
{ 1 , 2 , 3 , 4 }, and ∅
SUBSET

 This is defined to be the set of all subsets of a
given set, written as P(A).
Example: I = { 1 , 2 , 3 , 4 }
P (L) = { { 1 } , { 2 } , { 3 } , { 4 } , { 1 , 2 } , { 1 , 3 } , { 1 ,
4 }, { 2 , 3 } , { 2 , 4 } , { 3 , 4 } , { 1 , 2 , 3 } , { 1 , 2 , 4 },
{ 1 , 3 , 4 } , { 2 , 3 , 4 }, { 1 , 2 , 3 , 4 }, ∅ }
elements / subsets in the .
POWER SET

 Two sets are disjoint if they have no element in
common.
Example:
D = { consonants of the English alphabet }
K = { vowels of English alphabet }
Sets K and C are disjoint since they do not have
elements in common.
DISJOINT SETS

CARDINALITY OF
THE SET
The cardinality of a set is its
size. For a finite set, the
cardinality of a set is the number
of members it contains. In
symbolic notation the size of a set
S is written |S|.

AXIOM OF
EXTENSION
This states 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.

AXIOM OF
EXTENSION
Through the Axiom of Extension, sets can be
written not like this:
× L = { a , b , b , c , d , e , e , f , g , h , h , i , j }
But can be written like any of these:
 L = { a , b , c , d , e , f , g , h , i , j }
 L= { j , g , c , a , e , b , h , f , i , d }

The English logician John Venn (1834–1923)
developed diagrams, which we now refer to as Venn
diagrams, that can be used to illustrate sets and
relationships between sets. In a Venn diagram, the
universal set is represented by a rectangular region,
and subsets of the universal set are generally
represented by oval or circular regions drawn inside
the rectangle. In a Venn Diagram, the size of the
rectangle or circle is not a concern.
VENN
DIAGRAMS

UNION
The union of sets A and
B, denoted by A B, is
∪
the set consisting of all
elements that belong to
either A or B or both.

UNION
Example for union of sets:
M = {1, 2, 3, 4, 5, 6, 7,
11}
J = {2, 4, 6, 8, 10, 12}
M J = {1, 2, 3, 4, 5, 6, 7,
∪
8, 10, 11, 12}

INTERSECTION
The intersection of sets
A and B, denoted by
A B, is the set
∩
consisting of all
elements that belong
both A and B.

INTERSECTION
Example for intersection
of sets:
M = {1, 2, 3, 4, 5, 6, 7,
11}
J = {2, 4, 6, 8, 10, 12}
M J = {2, 4, 6}
∩

COMPLEMENT
The complement of set
A is defined as the set
consisting of all
elements in U that are
not in A.
𝐴={𝑥∈𝑈|𝑥𝐴}

COMPLEMENT
Example:
U = { x N | x < 21}
∈
J = {2, 4, 6, 8, 10, 12}
J' = { 1, 3, 5, 7, 9, 11, 13,
14, 15, 16, 17, 18,
19, 20}
𝐴={𝑥∈𝑈|𝑥𝐴}

DIFFERENCE
The difference or
relative complement of
two sets A and B,
denoted by A–B, is the
set consisting of all
elements in A that are
not in B.

DIFFERENCE
Example for difference of
sets:
M = {1, 2, 3, 4, 5, 6, 7, 11}
J = {2, 4, 6, 8, 10, 12}
M J = {1, 3, 5, 7, 11}
J M = {8, 10, 12 }

1
A 12>12 𝑥+11 𝑦
Write a negation for the
statement:
B
C12 𝑥 +11 𝑦 ≤ 12
D 12 𝑥 +11 𝑦 >11

2
A
It is not the case that students are happy or
teachers are not happy.
Translate the symbolic compound statement into
words. Let represent the statement: "Students
are happy" and let represent the statement:
"Teachers are happy."
B
Students are not happy and teachers are not
happy.
C It is not the case that students are happy and
teachers are not happy.
D
Students are not happy or teachers are not
happy.

3
A
We may not deduct points if there is no work
to justify your answer.
Write the negation of the statement: We will
deduct points if there is no work to justify your
answer. If not a statement, state so.
B
We will not deduct points if there is no work
to justify your answer.
C We will not deduct points if there is work to
justify your answer
D The excerpt is not a statement.

4
A Do not make sure that you fill in the circle on the
answer sheet that corresponds to your answer choice.
Write the negation of the statement: Make sure
that you fill in the circle on the answer sheet that
corresponds to your answer choice. If not a
statement, state so.
B
Do not make sure that you do not fill in the circle on the
C Make sure that you do not fill in the circle on the
D The excerpt is not a statement.

5
A 25
Give the number of rows in the truth
table for the compound statement:
B10
C
8
D
32

SOLUTION
We have 5 logical statements: p, q, r, s, and t.
25
=32

6
A 32
Find the number of subsets of the set:
B24
C
28
D
16

SOLUTION
We have 5 elements in the given set.
25
=32

7
A 6
Find the number of subsets of the set:
B8
C
256
D
1024

SOLUTION
We have 10 elements in the given set:
{18, 20, 22, 24, 26, 28, 30, 32, 34, 36}
210
= 1024

8 Construct a truth table for the
compound statement:
A.
s p (~s ∧~p)
T T F
T F T
F T T
F F T
B.
s p (~s ∧~p)
T T F
T F F
F T F
F F T D.
s p (~s ∧~p)
T T F
T F F
F T F
F F F
C.
s p (~s ∧~p)
T T T
T F F
F T F
F F T

SOLUTION
Construct a truth table for the compound statement:
T T F F F
T F F T F
F T T F F
F F T T T

9 Construct a truth table for the
compound statement:
A. B.
q s ~[~(q ∨s)] q s ~[~(q ∨s)]
T T T T T T
T F F T F T
F T T F T T
F F F F F F
C. D.
q s ~[~(q ∨s)] q s ~[~(q ∨s)]
T T F T T T
T F F T F T
F T F F T F
F F T F F F

SOLUTION
Construct a truth table for the compound statement:
T T T F T
T F T F T
F T T F T
F F F T F

10
Construct a truth table for the
compound statement:

SOLUTION
Construct a truth table for the statement:
)
T T T F T T
T F F F F T
F T T T T T
F F T T T T

10
Construct a truth table for the compound
statement:

11
Construct a truth table for the
compound statement:
A. B.
p q ~(p → q) → (p ∧~q) p q ~(p → q) → (p ∧~q)
T T T T T T
T F F T F T
F T F F T T
F F T F F T
C. D.
p q ~(p → q) → (p ∧~q) p q ~(p → q) → (p ∧~q)
T T T T T T
T F F T F F
F T F F T T
F F F F F T

SOLUTION
Construct a truth table for the statement:
T T T F F F T
T F F T T T T
F T T F F F T
F F T F T F T

12
A {t, v, x}
Let U = {q, r, s, t, u, v, w, x, y, z}; A = {q, s, u, w, y};
B = {q, s, y, z}; and C = {v, w, x, y, z}. List the
members of the indicated set, using set braces.
Find .
B {q,s,t,u,v,w,x,y}
C {r,t,u,v,w,x,z}
D{ s , u , w }

SOLUTION
Let U = {q, r, s, t, u, v, w, x, y, z}; A = {q, s, u, w, y}; B = {q, s, y, z}; and C
= {v, w, x, y, z}. List the members of the indicated set, using set
braces. Find .
U = {q, r, s, t, u, v, w, x, y, z}
A = {q, s, u, w, y}
B = {q, s, y, z}
𝐴∩ 𝐵={𝑞,𝑠 , 𝑦 }
( 𝐴∩𝐵)′={𝑟 ,𝑡 ,𝑢 ,𝑣 ,𝑤,𝑥 ,𝑧 }

13
A {r, t, u}
Let U = {q, r, s, t, u, v, w, x, y, z}; A = {q, s, u, w, y};
B = {q, s, y, z}; and C = {v, w, x, y, z}. List the
members of the indicated set, using set braces.
Find .
B {q,r,s,t,u,w}
C {q,r,s,t,u,v,w,x,y}
D{r , t , u, w }

SOLUTION
Let U = {q, r, s, t, u, v, w, x, y, z}; A = {q, s, u, w, y}; B = {q, s, y, z}; and C
= {v, w, x, y, z}. List the members of the indicated set, using set
braces. Find .
U = {q, r, s, t, u, v, w, x, y, z}
A = {q, s, u, w, y}
B = {q, s, y, z}
C = {v, w, x, y, z}
𝐵′={𝑟 ,𝑡 ,𝑢,𝑣 ,𝑤,𝑥}
𝐶′
={𝑞, 𝑟 , 𝑠 ,𝑡 ,𝑢}
𝐴∪𝐶 ′={𝑞,𝑟 ,𝑠,𝑡,𝑢,𝑤 , 𝑦 }
𝐵 ′ ∩( 𝐴∪𝐶′
)={𝑟 ,𝑡 ,𝑢,𝑤 }

14
A 11
, and ; what is ?
B12
C
13
D
10

SOLUTION
, and ; what is ?
𝑛( 𝐴∪ 𝐵)=𝑛( 𝐴)+𝑛(𝐵)−𝑛( 𝐴∩𝐵)
A B
2
2 7
𝑛( 𝐴∪ 𝐵)=4+9−2=11

15
Shade the Venn diagram to
represent the set: A' B'
∩
A
B
C
D

SOLUTION
A' B'
∩
A B
A'
A B
B'
A B

16
Shade the Venn diagram to
represent the set:
A
B
C
D

SOLUTION
( ) (𝐴 ∪ 𝐵)′
∩
∪ ∪′
A B
𝐴
∩ 𝐵
A B
𝐴
∪ 𝐵
A B
( )’
𝐴 ∪ 𝐵
A B

17
A 48
Use a Venn Diagram and the given information to
determine the number of elements in the
indicated region. If and , find .
B 4
C 56
D
12

SOLUTION
If ( ) = 60, ( ) = 34, ( ) = 22, and ( ) =
𝑛 𝑈 𝑛 𝐴 𝑛 𝐵 𝑛 𝐴 ∩ 𝐵
8, find ( ) .
𝑛 𝐴 ∪ 𝐵 ′
A B
8
26 14
12

18
A 7
If , , , , , , , and , Find .
B 9
C 10
D
8

SOLUTION
𝑛( ) = 77
𝐴 ∪ 𝐵 ∪ 𝐶
𝑛( ) = 11
𝐴 ∩ 𝐵 ∩ 𝐶
𝑛( ) = 24
𝐴 ∩ 𝐵
𝑛( ) = 21
𝐴 ∩ 𝐶
𝑛( ) = 19
𝐵 ∩ 𝐶
𝑛( ) = 56
𝐴
𝑛( ) = 38
𝐵
𝑛( ) = 36
𝐶
Find ( ).
𝑛 𝐴′ ∩ 𝐵 ∩ 𝐶
A
B
C
11
13
10
8
22
6
7

19
A 19
A survey of 240 families showed that 91 had a dog; 70
had a cat; 31 had a dog and a cat; 91 had neither a cat
nor a dog, and in addition did not have a parakeet; 7
had a cat, a dog, and a parakeet. How many had a
parakeet only?
B 24
C 34
D
29

SOLUTION
91 had a dog
70 had a cat
31 had a dog and a cat
91 had neither a cat
nor a dog, and in
addition did not have a
parakeet
7 had a cat, a dog, and
a parakeet.
C
D
P
7
24
91
60
3
9

20
A True
Determine whether the statement is
true or false.
B False

21
A True
Determine whether the statement is true
or false.
B False

22
A True
Determine whether the statement is true or
false.
B False

23
A True
Let A = {1, 3, 5, 7}; B = {5, 6, 7, 8}; C = {5, 8};
D = {2, 5, 8}; and U = {1, 2, 3, 4, 5, 6, 7, 8}.
Determine whether the given statement is
true or false.
B False

24
A True
Let A = {1, 3, 5, 7}; B = {5, 6, 7, 8}; C = {5, 8};
D = {2, 5, 8}; and U = {1, 2, 3, 4, 5, 6, 7, 8}.
true or false.
B False

25
A True
Let A = {1, 3, 5, 7}; B = {5, 6, 7, 8}; C = {5, 8};
D = {2, 5, 8}; and U = {1, 2, 3, 4, 5, 6, 7, 8}.
true or false.
B False

26
A True
Let A = {1, 3, 5, 7}; B = {5, 6, 7, 8}; C = {5, 8};
D = {2, 5, 8}; and U = {1, 2, 3, 4, 5, 6, 7, 8}.
true or false. C ⊈ B
B False

27
A True
Let A = {1, 3, 5, 7}; B = {5, 6, 7, 8}; C = {5, 8};
D = {2, 5, 8}; and U = {1, 2, 3, 4, 5, 6, 7, 8}.
true or false. A A
⊂
B False

28
A True
Let represent a true statement, and let
and represent false statements. Find the
truth value of the given compound
statement:
B False

SOLUTION
Find the truth value of the given compound statement:
~(~ ~ ) (~ ~ )
𝑝 ∧ 𝑞 ∨ 𝑟 ∨ 𝑝
T F F F T T F T T
T

29
A 𝑁𝑜𝑡 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡
Are the statements equivalent?
B 𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡
C 𝑀𝑎𝑦𝑏𝑒
D 𝑁𝑜 𝑎𝑛𝑠𝑤𝑒𝑟

SOLUTION
Are the statements ~( ) ~ equivalent?
𝑞 → 𝑝 𝑎𝑛𝑑 𝑞 ∧ 𝑝
T T T F F F
T F F T T F
F T T F F F
F F T F T F

30
A 𝑁𝑜𝑡 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡
Are the statements equivalent?
B 𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡
C 𝑀𝑎𝑦𝑏𝑒
D 𝑁𝑜 𝑎𝑛𝑠𝑤𝑒𝑟

SOLUTION
Are the statements ~ ~ ~ equivalent?
𝑞 ∧ 𝑝 𝑎𝑛𝑑 𝑝 → 𝑞
T T F F F T
T F T T F F
F T F F T T
F F T F T T

31
A True
Let p represent a true statement, and let q
represents a false statement. Find the truth
value of the compound statement:
B False

SOLUTION
Find the truth value of the compound statement:
T F F F T F T

32
A The sun does not come out tomorrow and the
roses will not open.
Write the statement, “If the sun comes out tomorrow,
the roses will open,” as an equivalent statement that
does not use the if . . . then connective. (Remember
that is equivalent to .)
B
The sun does not come out tomorrow or the roses
will not open.
C The sun does not come out tomorrow or the roses
will open.
D
The sun comes out tomorrow and the roses will
not open.

33
A If it is raining, you do not take your umbrella.
Write the negation of the statement: If it is raining,
you take your umbrella. (Remember that the
negation of p q is p ~q.)
→ ∧
B
It is raining and you do not take your
umbrella.
C
It is not raining and you do not take your
umbrella.
D It is not raining and you take your umbrella.

34
A 7x + 2y > -3, so the answer is not "Lake ".
Write the negation of the statement: If
7x + 2y > 3, the answer is "Lake ".
B 7x + 2y ≤ 3 and the answer is not "Lake ".
C If 7x + 2y > 3, the answer is not "Lake ".
D 7x + 2y > 3 and the answer is not "Lake ".

35
A Collection of objects
A set is:
B Well-defined collection of objects
C Elements of the set
D Description

36
A 106
In a survey of university students, 64 had taken mathematics
course, 94 had taken chemistry course, 58 had taken physics
course, 28 had taken mathematics and physics, 26 had taken
mathematics and chemistry, 22 had taken chemistry and
physics course, and 14 had taken all the three courses. Find
how many had taken one course only.
B 94
C 82
D
72

SOLUTION
64 had taken math
94 had taken chemistry
58 had taken physics
28 had taken math &
physics
26 had taken math &
chem
22 had taken chem &
physics
14 had taken all
C
P
M
1
4
8
12
14
22
60 24

37
A 10
An advertising agency finds that, of its 170 clients, 115
use Television, 110 use Radio and 130 use Magazines.
Also 85 use Television and Magazines, 75 use
Television and Radio, 95 use Radio and Magazines, 70
use all the three. Draw Venn diagram to represent
these data. How many uses only Radio?
B 25
C 15
D
50

SOLUTION
115 use Television
110 use Radio
130 use Magazines
85 use TV & Magazines
75 use TV & Radio
95 use Radio &
Magazines
70 use all the three
T
R
M
7
0
5
15
25
10
25 20

38
A 10
An advertising agency finds that, of its 170 clients, 115
use Television, 110 use Radio and 130 use Magazines.
Also 85 use Television and Magazines, 75 use
Television and Radio, 95 use Radio and Magazines, 70
use all the three. Draw Venn diagram to represent
these data. How many uses only Television?
B 25
C 15
D
50

39
A 10
An advertising agency finds that, of its 170 clients, 115 use
Television, 110 use Radio and 130 use Magazines. Also 85
use Television and Magazines, 75 use Television and Radio,
95 use Radio and Magazines, 70 use all the three. Draw Venn
diagram to represent these data. How many uses both
Television and Magazine but not radio?
B 25
C 15
D
50

40
A 8
There are 30 students in a class. Among them, 8
students are learning both English and French. A
total of 18 students are learning English. If every
student is learning at least one language, how
many students are learning French in total?
B 12
C 20
D
28

SOLUTION
30 students in a class
8 students are learning
both English and
French
18 students are learning
English
every student is learning
at least one language
E F
8
10 12

MATHEMATICS LOGIC AND SET THEORY PRESENTATION.pptx

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