TITLE: Why is "A only if B" equivalent to "(not A) or B"?
QUESTION [11 upvotes]: I've encountered this recently and I just can't wrap my head around it. My book states that 
$$A \rightarrow B \equiv \neg A \lor B$$
It's my understanding that $A \rightarrow B$ means that if $A$ is true, then $B$ is true. 
But, $\neg A \lor B$ would allow for us to have that $\neg A$ is true and $B$ is true, which seems to be pretty much that exact opposite of $A \rightarrow B$, so how can these be equivalent?

REPLY [5 votes]: Here is a table showing all possible value-combinations for A and B in the first two columns. You can see that columns 3 and 5 are the same, therefore both predicates must be equivalent.  
| A | B | A->B | not A | (not A) or B |
|---|---|------|-------|--------------|
| T | T | T    | F     | T            |
| F | T | T    | T     | T            |
| T | F | F    | F     | F            |
| F | F | T    | T     | T            |