Logical Equivalence
Two statements are logically equivalent if they imply one another; contrapositive of S = S; inverse of S = converse of S
In logic, logical equivalence is a relationship between two statements that have the same truth value in all possible circumstances. That is, two statements are logically equivalent if they have the same meaning and truth value. Logical equivalence is also known as propositional equivalence or truth-functional equivalence.
To determine if two statements are logically equivalent, we can construct a truth table for each statement and compare the results. If the truth table for each statement is identical, then the statements are logically equivalent.
For example, consider the statements p and q and q and p. We can construct a truth table for each statement:
| p | q | p and q |
|——–|——–|———|
| true | true | true |
| true | false | false |
| false | true | false |
| false | false | false |
| p | q | q and p |
|——–|——–|———|
| true | true | true |
| true | false | false |
| false | true | false |
| false | false | false |
As we can see, the truth table for p and q and q and p are identical. Therefore, they are logically equivalent.
Logical equivalence is important in logic and mathematics because it allows us to simplify complex statements and arguments. By identifying logically equivalent statements, we can replace them with simpler or more intuitive statements without changing the meaning of the argument.
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