Biconditional Statement

Question

Upon reading my textbook it gives a definition for a biconditional statement as the following:

Given statement variables $p$ and $q$, the biconditional of $p$ and $q$ is "$p$ if, and only if, $q$ and is denoted $p \leftrightarrow q$.

It then mentions that:

It is true if both $p$ and $q$ have the same truth values and is false if $p$ and $q$ have opposite truth values. The words...

My question is, from the second part of the definition, is it talking about if $p$ is equal to true and $q$ is equal to true then the outcome will be true? Likewise if both are false, the outcome will be true? Also, if that is the case, is it saying that the rest, i.e. $p$ being true or false and $q$ being its opposite, the outcome will always be false?

Answer 1

Yes, your interpretation is correct. Here is the truth table for the biconditional:

$$\begin{array}{c|c|c} p & q & p\iff q \\\hline T & T & T\\\hline T & F & F \\\hline F & T & F \\\hline F & F & T \end{array}$$

Answer 2

You are right. See the following truth table:

Answer 3

The biconditional "p if and only if q" can be read as "p implies q AND q implies p". In this way analyzing a biconditional reduces to analyzing two implications.