Introduction to Boolean Algebra
 Motivated by electronic circuits
 Boolean algebra is a set of operators applicable to elements
from the set {0, 1}:
 complement: 0 = 1, 1 = 0
 Boolean product: 1  1 = 1, 1  0 = 0, 0  1 = 0, 0  0 = 0
 Boolean sum: 1 + 1 = 1, 1 + 0 = 1, 0 + 1 = 1, 0 + 0 = 0
 Example: Find the value of 1  0 + (0 + 1)
Solution : 1  0 + (0 + 1) = 0 + 1
=0+0
=0
Boolean Expressions and Functions
 Let B={0,1}. A Boolean variable x has only values from B
 A Boolean expression is built up recursively from 0, 1,
Boolean variables/expressions, using Boolean operators
 Bn = {(x1,x2,…,xn) | xi ∈ B for 1≤ i ≤ n } is the set of all
possible n-tuples of 0s and 1s
 Boolean function F of degree n is a mapping from Bn to B
 F can be represented by a Boolean expression
 Example: F(x, y) = xy + 𝑥𝑦 from
the set of ordered pairs of Boolean
variables to the set {0, 1} is a
Boolean function of degree 2
Boolean Expressions and Functions
 Example: Find the values of the Boolean function
F(x, y, z) = xy + 𝑧
Boolean Functions
𝒏
𝟐
𝟐
There are
different Boolean
functions of degree n because:
 with n variables there are 2n
different n-tuples of 0s and 1s
 each tuple is assigned 0 or 1
Example: there are 16 Boolean functions of degree 2
Identities
 Boolean identities
correspond to identities
of propositional logic
and of set identities
 All identities (except
the first) come in pairs.
 Each element of the
pair is the dual of the
other
Identities of Boolean Algebra

Identities can be proven using tables analogous to truth
tables of logic
 Example: Prove the distributive law x(y+z) = xy+xz