1. Boolean Functions

2. 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, = 0
Example: Find the value of 1  0 + (0+1)
Solution : 1  (0+1) =
= 0 + 0
= 0

3. 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 + 𝑥 𝑦 from the set of ordered pairs of Boolean variables to the set {0, 1} is a Boolean function of degree 2

4. Boolean Expressions and Functions
Example: Find the values of the Boolean function F(x, y, z) = xy + 𝑧

5. Boolean Functions
There are 𝟐 𝟐 𝒏 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

6. 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

7. 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