Boolean expressions of two variables

Since any two boolean expressions with the same truth table are equivalent we can enumerate all the boolean expressions of a given number of variables. For instance with two variables, a and b there are four different combinations of possible values (i.e. 2²) In other words, if we wrote out the truth table for an expression involving those two variables we would need four rows. For instance here's the truth table for a && b:

a	b	a && b
false	false	false
false	true	false
true	false	false
true	true	true

And since the value in each row of the final column is either true or false, there are only 2⁴ possible unique truth tables. Thus any boolean expression of two variables must be equivalent to one of just sixteen distinct boolean expressions.

The table below shows the canonical form of those sixteen expressions with the corresponding final column from the truth table flipped into a row. If applicable, the “DeMorgan'd” column contains the equivalent obtained by applying one of DeMorgan’s Laws. Note also that some of the expressions don't actually use both variables; two of them don't use them at all! But these are all the possible truth tables for two-variable expressions. Thus if you ever have a boolean expression of just two variables that is more complicated than the expressions here, you can definitely simplify it as it must be equivalent to one of these.

Expression	DeMorgan’d	a = false b = false	a = false b = true	a = true b = false	a = true b = true	Name
`false`		false	false	false	false	Contradiction
`a && b`	`!(!a \|\| !b)`	false	false	false	true	And
`a && !b`	`!(!a \|\| b)`	false	false	true	false
`a`		false	false	true	true
`!a && b`	`!(a \|\| !b)`	false	true	false	false
`b`		false	true	false	true
`a != b`	`!(a == b)`	false	true	true	false	XOR (Exclusive Or)
`a \|\| b`	`!(!a && !b)`	false	true	true	true	Or
`!(a \|\| b)`	`!a && !b`	true	false	false	false	NOR (Not Or)^*
`a == b`	`!(a != b)`	true	false	false	true	Logical Equality or XNOR (Exclusive NOR)
`!b`		true	false	true	false
`a \|\| !b`	`!(!a && b)`	true	false	true	true
`!a`		true	true	false	false
`!a \|\| b`	`!(a && !b)`	true	true	false	true
`!(a && b)`	`!a \|\| !b`	true	true	true	false	NAND (Not AND)^*
`true`		true	true	true	true	Tautology

^*Fun fact about NAND and NOR: all boolean operators can be built from some combination of NANDs or from some combination of NORs. In digital circuits (which are built out of “logic gates” that correspond to the basic boolean operators) this means that any computer circuit can be built entirely out of exclusively NAND gates wired together into an appropriate configuration or exclusively from NOR gates.