Logical Equivalence - Notesformsc

We have seen prepositions, connectives and compound prepositions; all possible combinations of truth values of the individual prepositions in a compound preposition is depicted in a truth-table. However, it is possible that another preposition or compound preposition has the same truth values in the truth table. This is called logical equivalence of two prepositions.

Verifying Logical Equivalence using Truth-Table

As we mentioned earlier, the simplest way to verify logical equivalence of two preposition or compound preposition is to create a truth table and compare the output of each logical expression.

The number of variables used in the truth-table for each expression may be different, but we shall only take variables that are common to both the expressions.

Example #1

Suppose you are asked to verify the logical equivalence of following expressions.

p \implies q \equiv \neg p \vee q

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The number of unique variables is $p, q$ . Therefore, our truth-table will contain $2^{2} = 4$ rows. The truth-table will contain truth values for all the expressions as given below.

$p$	$q$	$\neg p$	$p \implies q$	$\neg p \vee q$
$T$	$T$	$F$	$T$	$T$
$T$	$F$	$F$	$F$	$F$
$F$	$T$	$T$	$T$	$T$
$F$	$F$	$T$	$T$	$T$

Truth-table to verify logical equivalence : $p \implies q \equiv \neg p \vee q$

The above table clearly shows that the logically both expressions give the same truth value for all input combinations, that is, $p \implies q \equiv \neg p \vee q$ is equivalent. Let us see some more examples of logical equivalences.

Example #2

In the second example, we will try to prove the logical equivalence of biconditional connective using truth table.

p \iff q \equiv p \implies q \wedge q \implies p

There are exactly two unique variables in above expressions. Therefore, the truth-table will contain 4 rows.

$p$	$q$	$p \iff q$	$p \implies q$	$q \implies p$	$p \implies q \wedge q \implies p$
$T$	$T$	$T$	$T$	$T$	$T$
$T$	$F$	$F$	$F$	$T$	$F$
$F$	$T$	$F$	$T$	$F$	$F$
$F$	$F$	$F$	$T$	$T$	$T$

Truth table for logical equivalence p<->q <=> p -> q and q -> p

The truth value for each row is same for both the expressions; therefore, both expressions are equivalent. There are several other logical expressions that we are going to explore in future lessons.

Verifying Logical Equivalence using Truth-Table

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