3.4
Equivalent Statements
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Equivalent Statements

Two statements are equivalent if both
statements have exactly the same truth values
in the answer columns of the truth tables.
 Symbols:  or 

In a truth table, if the answer columns are
identical, the statements are equivalent.
If the answer columns are not identical, the
statements are not equivalent.

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De Morgan’s Laws
~ (p  q)  ~ p  ~ q
~ (p  q)  ~ p  ~ q
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Example: Using De Morgan’s Laws to
Write an Equivalent Statement
Use De Morgan’s laws to write a statement
logically equivalent to “Benjamin Franklin was
not a U.S. president, but he signed the
Declaration of Independence.”
Solution: Let
p: Benjamin Franklin was a U.S. president
q: Benjamin Franklin signed the Declaration of
Independence
The statement symbolically is ~p Λ q.
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Example: Using De Morgan’s Laws to
Write an Equivalent Statement
The logically equivalent statement in symbolic
form is

~ p  q ~ p  ~ q

Therefore, the logically equivalent statement to
the given statement is:
“It is false that Benjamin Franklin was a U.S.
president or Benjamin Franklin did not sign the
Declaration of Independence.”
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p  q  ~p  q


To change a conditional statement into a
disjunction, negate the antecedent, change the
conditional symbol to a disjunction symbol, and
keep the consequent the same.
To change a disjunction statement to a
conditional statement, negate the first
statement, change the disjunction symbol to a
conditional symbol, and keep the second
statement the same.
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Variations of the Conditional Statement

The variations of conditional statements are the
converse of the conditional, the inverse of the
conditional, and the contrapositive of the
conditional.
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Variations of the Conditional Statement
Name
Conditional
Converse of the
conditional
Inverse of the
conditional
Contrapositive of
the conditional
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Symbolic
Form
p  q
q

Read
“if p, then q”
“if q, then p”
p
~p

~q “if not p, then not q”
~q

~p
“if not q, then not p”
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Homework

P. 136 # 9 – 71 eoo
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