Computer Class notes

Boolean algebra

A branch of mathematics dealing with variables that can have only two possible values, typically 0 and 1, or false and true.

Boolean Variables

Represented by letters (e.g., A, B, X) and can only take on the values 0 (False) or 1 (True).

Literals

A Boolean variable or its complement (e.g., A or A̅).

Boolean Expression

A combination of Boolean variables and operators.

AND Operation

The output is 1 if and only if all inputs are 1.

Symbol for AND Operation

Bullets (·) or no symbol (e.g., A·B or AB).

OR Operation

The output is 1 if at least one input is 1.

Symbol for OR Operation

Plus sign (+) (e.g., A + B).

NOT Operation

Inverts the input; if the input is 0, the output is 1, and vice versa.

Symbol for NOT Operation

A bar over the variable (A̅) or prime (A').

Commutative Law

States that A + B = B + A and A·B = B·A.

Associative Law

States that (A + B) + C = A + (B + C) and (A·B)·C = A·(B·C).

Distributive Law

States that A·(B + C) = (A·B) + (A·C) and A + (B·C) = (A + B)·(A + C).

Identity Law

States that A + 0 = A and A·1 = A.

Complement Law

States that A + A̅ = 1 and A·A̅ = 0.

Idempotent Law

States that A + A = A and A·A = A.

Null Law

States that A + 1 = 1 and A·0 = 0.

Involution Law

States that A̅̅ = A.

Absorption Law

States that A + (A·B) = A and A·(A + B) = A.

De Morgan's Laws

States that A̅ + B̅ = (A·B)̅ and A̅·B̅ = (A + B)̅.

Logic Gates

Electronic circuits that implement Boolean operations, with each basic operation corresponding to a specific gate.

AND Gate

Implements the AND operation.

OR Gate

Implements the OR operation.

NOT Gate

Implements the NOT operation.

XOR Gate

A gate that implements the Exclusive OR operation.

XNOR Gate

A gate that implements the Exclusive NOR operation.