Boolean Algebra

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# Boolean Algebra ## 9.1 Boolean Algebra Axioms A Boolean algebra is a set $B$ with operations $+$ (OR), $\cdot$ (AND), $'$ (NOT), and elements $0, 1$ satisfying: Axiom AND form OR form Commutative $a \cdot b = b \cdot a$ $a + b = b + a$ Distributive $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$ $a + (b \cdot c) = (...

Boolean Algebra

9.1 Boolean Algebra Axioms

A Boolean algebra is a set

B

with operations

+

(OR),

\cdot

(AND),

'

(NOT), and elements

0, 1

satisfying:

Axiom	AND form	OR form
Commutative	$a \cdot b = b \cdot a$	$a + b = b + a$
Distributive	$a \cdot (b + c) = (a \cdot b) + (a \cdot c)$	$a + (b \cdot c) = (a + b) \cdot (a + c)$
Identity	$a \cdot 1 = a$	$a + 0 = a$
Complement	$a \cdot a' = 0$	$a + a' = 1$

9.2 Boolean Functions

A Boolean function

f: \{0,1\}^n \to \{0,1\}

can be expressed in:

DNF (Sum of Products): OR of ANDs
CNF (Product of Sums): AND of ORs

Minterms

A minterm is an AND of all variables (complemented or not). Any Boolean function can be expressed as a sum of minterms.

9.3 Karnaugh Maps

A K-map is a graphical method for minimizing Boolean expressions. For 2 variables:

\begin{array}{c|cc} & \bar{y} & y \\ \hline \bar{x} & 0 & 1 \\ x & 1 & 1 \end{array}

This represents

x + \bar{y}

✅ Practice Questions

Q1: Simplify

F(x,y,z) = x'y'z + x'yz + xy'z' + xy'z

Solution

$= x'z(y' + y) + xy'(z' + z) = x'z(1) + xy'(1) = x'z + xy'$ Q2: Prove $a + a'b = a + b$ . Solution

$a + a'b = (a + a')(a + b) = 1(a + b) = a + b$ (using distributivity). Join Discord PreviousNumber Theory & RSANextProbability Applications

What this note is really saying

Boolean Algebra

9.1 Boolean Algebra Axioms

9.2 Boolean Functions

Minterms

9.3 Karnaugh Maps

✅ Practice Questions

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