Chapter 6
Logic
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6.1
Statements
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Symbolic Logic
Symbolic logic is to use formal mathematics
with symbols to represent statements and
arguments in very day life.
 It is a Time-saver om argumentation.


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It helps prevent logical confusion when
dealing with complex argument.
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• Statement

A statement is a declarative sentence that is
either true or false, but not both
simultaneously.

Statements that involve one or more of the
connectives ``and'', ``or'', ``not'', ``if then'' and
`` if and only if '' are compound statements
(otherwise they are simple statements).
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Example 1 and Your Turn 1
Decide whether each statement is compound.
(a) George Washington was the first U.S. president, and John
Adams was his vice president.
(b) If what you’ve told me is true, then we are in great peril.
(c) We drove across New Mexico toward the town with the curious
name Truth or Consequences.
(d) The money is not there.
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Your Turn 1: “I bought Ben and Jerry’s ice cream.”
Solution: This statement is not compound. Even though and is a
connective, here it is to connect two people Ben and Jerry.
It is not connecting two statements.
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Negation

The negation of a statement is the
contradiction or denial of something.

The negation of “I play piano” is “I do not
play piano.”

It is equivalent to say “ It is not true that I
play piano”.
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Example 2 and Your Turn 2
Give the negation of each statement.
(a) California is the most populous state in the country.
(b) It is not raining today.
Your Turn 2: “Wal-Mart is not the largest corporation in the USA.”
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Your Turn 3
Write the negation of the following inequality.
4x  2 y  5
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Your Turn 4
Let h represent “My backpack is heavy,” and r represent “It’s
going to rain.”
Write the symbolic statement in words h   r.
Solution:
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Truth Value and Truth Table

The truth value of a statement is T if it is true
and F if it is false.

The truth value of a compound statement is
determined from the truth values of its simple
components under certain rules.
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Conjunction

The conjunction is the word logicians use for
p and q, denoted p^q.
Ex: My birthday is in October and yours in in
Novememter.
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It is only true when both component
statements are true.
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Your Turn 5
Let p represent “ 7 < 2 ” and q represent “ 4 > 3”. Find the truth
value of  p  q.
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It is only false when both component statements are
false.
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Your Turn 6
If p is false, q is true, and r is false, find the truth value of the
statement: ( p  q)  r.
Solution:
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Your Turn 7
Let p represent “7 < 2, ” q represent “4>3,” and r represent
“ 2 > 8.” Find the truth value of p  ( q  r ).
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6.2
Truth Tables and
Equivalent Statements
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Ex: Construct a true table for (~pq) ~q
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Your Turn 1
Construct a truth table for p  ( p  q).
If p and q are both true, find the truth value of p  ( p  q).
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Your Turn 2
Construct a truth table for the statement: “I do not order pizza,
or you do not make dinner and I order pizza.”
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Ex: Construct the truth table for (~pq) ~q
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Your Turn 3
Find the negation of the statement: “You do not make dinner or
I order pizza.”
and I do not order pizza.
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6.3
The Conditional and
Circuits
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Conditionals

A conditional statement is a compound
statement that uses the connective
if….then,
or anything equivalent.
Ex: If it rains, then I carry my umbrella.
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The conditional is written with an arrow, so
that “if p, then q” is symbolized as “p → q”,
read as “ p implies q”.
 p is the antecedent.
q is the consequent.


Ex: Winners never quit.
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Ex: If you eat Wheat Crunchies, then
you will be full of energy.
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Your Turn 1
Find the truth value of the statement, “If Little Rock is
the capital of Arkansas, then New York City is the capital of
New York.”
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Your Turn 2
Determine if the statement is true or false: 4  5  F .
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Your Turn 3
If p, q, and r are all false, find the truth value of the statement
 q  ( p  r ).
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Construct a truth table for each statement
Ex: ( ~p → ~𝑞) ~(𝑝
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~ 𝑞)
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Example: Write the following statement
without using if…then

If you do the homework, then you will pass
the quiz.
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Example: Write the negation of each
statement.

(a)If you go to the left, I’ll go to the right.

(b) It must be alive if it is breathing.
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6.4
More on the
Conditional
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Example: Equivalent Statements

Write the following statement in eight
different equivalent ways, using the common
translations of p → 𝑞 in the box above.
If you answer this survey, then you will be
entered in the drawing.
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Example 2 Equivalent Statement

Write each statement in the form “if p, then q”.
(a) Possession of a valid identification card is
necessary for admission.
(b) You should use this door only if there is an
emergency.
(c) All who are weary can come and rest.
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Example: Related Conditional Statement
If Cauchy is a cat, then Cauchy is a mammal.
(a) The converse

(b) The inverse
(c) The contrapositive
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Biconditionals
In logic, a biconditional is a compound
statement formed by combining two
conditionals under "and."
 A biconditional is read as "[some fact] if and
only if [another fact]" and is true when
the truth values of both facts are exactly the
same - BOTH TRUE or BOTH FALSE.
 Biconditionals are true when both
statements (facts) have the exact same truth
value.

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Example
Tell whether each statement is true or false.
(a) A triangle is isosceles if and only if the
tringle has two congruent sides.

(b)
Alaska is one of the original 13 states if and
only if kangaroos can fly.
(c)
2+2=4 if and only if 7>10
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