Circuits, Truth Tables & Boolean Algebra

Expressions Can describe circuits in terms of Boolean expression

Expressions Write Q as a function of A, B and C

Expressions Write Q as a function of A, B and C

Expressions Write Q as a function of A, B and C

Expressions Write Q as a function of A, B and C

Expressions Can represent a circuit's output with Boolean expression: AB + AC

Sum Of Products Can convert any truth table to expression Identify ways to get 1 output X = 1 if red OR green OR yellow

Sum Of Products Sum of Products : Individual terms are products Terms OR’d together X = A B + A B + AB

Multiple Inputs Each product shows EVERY input Either raw or inverted Ex: Each product has A or A B or B C or C

SOP Circuit Can build circuit from SOP expression: X = A B + A B + AB Inputs feed layer of AND gates Some inputs inverted first OR gate takes ANDs X = A B + A B + AB

Multiple Outputs Multiple outputs Each output separate function/circuit IN OUT A B X Y 1

Multiple Outputs Multiple outputs X = AB Each output separate function/circuit X = AB IN OUT A B X Y 1

Multiple Outputs Multiple outputs X = AB Y = AB + A B Each output separate function/circuit X = AB Y = AB + A B IN OUT A B X Y 1

Logisim Build simplified circuit from truth table… Window  Combinational Analysis Build truth table for circuit… Project  Analyze Circuit

Multiple Outputs Multiple outputs X = AB Y = AB + A B Each output separate function/circuit X = AB Y = AB + A B

SOP Circuit Fun Fact Can build circuit from SOP expression… … but can often do better…

Expressions AB + AC A(B + C) Fun Fact We can simplify Boolean expressions AB + AC A(B + C)

Fun Fact Simplification Thus we can simplify circuits AB + AC A(B + C)

Simplification Fun Fact May be able to simplify SOP X = A B + A B + AB = A B + A( B + B) = A B + A(1) = A B + A = B + A = A + B

Single Gate Logic Fun Fact Can make NOT, AND & OR with only NAND

Single Gate Logic Fun Fact Can build circuit from SOP expression: Any circuit can be build with NOT, AND & OR

Fun Fact Single Gate Logic Replace NOT with NAND

Single Gate Logic Fun Fact Put inversion bubbles out of AND & into OR 2 negations cancel out

Single Gate Logic Fun Fact Convert A + B + …. to AB... De Morgan's Theorem

Single Gate Logic Fun Fact Circuit built entirely with NAND Can do the same trick with NOR