These notes are useful to people just starting out in formal mathematics and logic. You might want to have them to hand whilst thinking about problems such as IFFY logic or Mind Your Ps and Qs.

At a basic level proof is based on the concepts of:

IF (something is true) THEN (something else is true)

There are three versions of this of this

(p is true) ONLY IF (q is true) [we say q is NECESSARY for p]

(note: in this case p cannot be true if q is false)

(p is true) IF (q is true) [we say q is SUFFICIENT for p]

(note: in this case p might be true if q is false)

(p is true) IF AND ONLY IF (q is true) [we say q is NECESSARY AND SUFFICIENT for p]

(note: in this case p and q are either both true or both false. They are logically equivalent)

Logical symbols

Statements in mathematical logic are sentences which are either true or false.

We can link statements using AND, OR and NOT

(p AND q) is true if and only if BOTH p and q are true

(p OR q) is true if and only if at least one of p or q is true

NOT(p) is true if and only if p is false.

Implication arrows $\Leftarrow, \Rightarrow, \Leftrightarrow$

These show the direction of the IF ... THEN logic as follows.

$p\Rightarrow q$ read as p implies q essentially means if p is true then q is also true

$p\Leftarrow q$ read as p is implied by q essentially means if q is true then p is also true

$p\Leftrightarrow q$ means $p\Rightarrow q$ AND $p\Leftarrow q$.