Logical Statements and Conditions

Implication and converse

An implication P \Rightarrow Q reads "if P then Q" — we say "P implies Q". It promises that whenever P is true, Q must be true as well.

P \Rightarrow Q \quad\equiv\quad \text{if } P \text{ then } Q

Swapping the two sides gives the converse Q \Rightarrow P. The converse is a different statement — it is not automatically true just because the original is. When both directions hold we write P \Leftrightarrow Q, read "P if and only if Q".

For example, x = 2 \Rightarrow x^2 = 4 is true. But its converse x^2 = 4 \Rightarrow x = 2 is false: from x^2 = 4 we could also have x = -2. So an implication can hold while its converse fails.

Necessary and sufficient

These two words name the two directions of an implication.

P is a sufficient condition for Q when P \Rightarrow Q — having P is enough to guarantee Q.
P is a necessary condition for Q when Q \Rightarrow P — you cannot have Q without P.

If P \Leftrightarrow Q, then P is both necessary and sufficient for Q — the two statements stand or fall together.

For statements P and Q:

P \Rightarrow Q means "if P then Q";
the converse Q \Rightarrow P may be false;
P \Leftrightarrow Q means both ways hold;
P is sufficient for Q when P \Rightarrow Q, and necessary when Q \Rightarrow P;
P \Leftrightarrow Q makes P necessary and sufficient for Q.