I'm reading Susanna Epps book on discrete mathematics and I have a question about the notation of logical equivalence.
In the books definition: "Two statement forms are called logically equivalent if, and only if, they have identical truth values for each possible substitution of statements for their statement variables… denoted $P\equiv Q$.
Two statements are called logically equivalent if, and only if , they have logically equivalent forms when identical component statement variables are used to replace identical component statements."
Later in the exercise section she writes: $p="x>5"$. Do you not use $\equiv$ for statement definitions and if so, how do you symbolise equivalence between two statements, $p\equiv q$ or $p=q$.
Since p and q by them selves could technically be seen as statement forms, is there a difference between $p\equiv q$ and $p=q$?
Best Answer
$p=q$ means the two statements are identical, i.e. that they are one single statement.
$p\equiv q$ means that the two statements are equivalent, but not (necessarily) identical.
For example, if $p$ is the statement "$x>5$", while $q$ is the statement "$\neg (x\leq 5)$", then the two statements are equivalent, but they are not identical.