$\require{cancel}$

Equations, Truth Tables and Circuits.

• These are equivalent.
• Starting with $f(a,b) = ab + a\overline{b}$ can you
1. Create the circuit?
2. Create the truth table?
• The circuit
• And has higher precedence than or.
• Truth Table
• $A$$B$$AB$ $\overline{B}$$A\overline{B}$ $AB + A\overline{B}$
0 0 0 1 00
0 1 0 0 00
1 0 0 1 11
1 1 1 0 01
• Wow, this looks like just A.
• On page B-696 they give us several laws of boolean algebra.
•  $f(A,B)$ $= AB + A\overline{B}$ $= A(B + \overline{B})$ $= A(1)$ $= A$
• We need to be careful of DeMorgan's law.
• $\overline{AB} = \overline{A} + \overline{B}$
• $\overline{A + B} = \overline{A} \cdot \overline{B}$
• Pick one, prove it by truth table, prove it by circuit.
• Can you translate this circuit into a Truth Table and a Boolean Expression?
• By the way, this is exclusive or (xor) $C \oplus D$
• Can you translate this truth table into the other two?
E F f(E,F)
0 0 1
0 1 0
1 0 0
1 1 1
• If you wish, the circuits are in equations.v
• More than two variables. Do the same thing.
• G H I f(G,H,I)
0 0 0 0
0 0 1 1 ($\overline{G}\overline{H}I$)
0 1 0 0
0 1 1 0
1 0 0 1 ($G\overline{H}\overline{I}$)
1 0 1 1 ($E\overline{H}I$)
1 1 0 0
1 1 1 1 ($GHI$)