Boolean algebra; what does <-> mean?

Question

Expression :$$(p\rightarrow q)\leftrightarrow(\neg q\rightarrow \neg p)$$ What does the symbol $\leftrightarrow$ mean ? Please explain by drawing the truth table for this expression and also with other examples if possible. I'm in a desperate situation so I'd really appreciate a quick response !

Answer 1

The expression $A\iff B$ means if and only if.

For $\iff$, we get the following truth table $$\begin{array}{c|c|c|} & B\text{ is true}& B\text{ is false}\\\hline A\text{ is true} & \text{true} & \text{false}\\\hline A\text{ is false} & \text{false} & \text{true}\\\hline \end{array}$$

Answer 2

$$A\leftrightarrow B$$ is the same as

$$(A\rightarrow B) \land (B\rightarrow A)$$

Answer 3

Symbol $\leftrightarrow$ or $\iff$ denote usually the equivalence, commonly known also as "NXOR", "if and only if" or "iff" for short (see also its Wikipedia page). More precisely $p \leftrightarrow q$ is equal to $$(p \to q) \land (q \to p)$$ or $$(p \land q) \lor (\neg p \land \neg q),$$ and the truth table is: $$ \begin{array}{cc|c} p & q & p \leftrightarrow q \\\hline 0 & 0 & 1\\ 0 & 1 & 0\\ 1 & 0 & 0\\ 1 & 1 & 1 \end{array} $$

I hope this helps $\ddot\smile$

Answer 4

All the answers above are great, and should help you. I will show you how I would do the full truth table for the logical statement: $$(p \Rightarrow q)\Longleftrightarrow(\neg q \Rightarrow \neg p)$$

\begin{array}{cc|c|c|c|c} p & q & \neg q & \neg p & \neg q \Rightarrow \neg p & p \Rightarrow q\\\hline T & T & F & F & T & T\\ T & F & T & F & F & F\\ F & T & F & T & T & T\\ F & F & T & T & T & T \end{array}