Logical Operators

Conjunction and Disjunction

AND, Conjunction

Definition

, logical AND, like set intersection

OR, Disjunction

Definition

, logical OR, like set set union

Logical Operator and Algebra

Properties

For all , we have:

Property	Explanation
	closed under addition
	closed under multiplication
	additive identity
	multiplicative identity
	addition is commutative
	multiplication is commutative
	addition is distributive
	multiplication is distributive
	addition is associative
	multiplication is associative
For all , there exists such that	additive inverse
For all , there exists such that	multiplicative inverse
There exists such that	at least 2 unique elements

De Morgan's Laws

Law

When expanding negation, the elements take the negation, and conjunction changes to disjunction and vice versa

For all ,

Remark

De Morgan's Laws work for an arbitrary number of variables:

Absorption Law

Law

For all ,

Proof

Proof of (1):

Proof for (2):

Partial Distributive Law

Law

For all ,

Intuition

This is just distributive law, but the first term gets lost

Proof

Proof for (1):

Proof (2):

Combining Law

Law

For all

Proof

Proof for (1):

Proof for (2):

Consensus Law

Law

For all

Proof

Proof for (1):

Proof for (2):

Tip

A good way to remember this law is, if you can rearrange the elements into a "chain" with the first and last variables being complements of each other, then you can KILL the middle element.

I.e