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

Expressions We can simplify boolean expressions AB + AC A(B + C)

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

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

Implication 1 Can build circuit from SOP expression: Any circuit can be build with NOT, AND & OR

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

Simplification Can simplify things… X = A B + A B + AB = A B + A( B + B) = A B + A(1) = A B + A = B + A = A + B

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

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

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

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

Single Gate Logic Fun Fact Any circuit can be build with NAND gates only…

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