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Digital Circuits

OR

AND

NOT ~p

p∨q

p∧qp and q

not p

pq

p

p

q p or q

p⋅q

p+q

′ p

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Textbook - Example 1.7

pq

(p⋅q ′ )

Textbook - Example 1.8

p

r

q (p+q)⋅ ′ r

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Textbook - Example 1.9

p

r

q

( ′ p ⋅q) +(p⋅ ′ r ′ )

001 011 010

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Textbook – Example 1.13

+ 0 1

0 00 01

1 01 10

Half-adder adds two binary numbers:

p q d0 d11 1 0 1

1 0 1 0

0 1 1 0

0 0 0 0

d0 ≡p ′ q + ′ p q

d1≡p•q

is equivalent to:

d0 ≡(p+q) ⋅(pq ′ )

001 011 010

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d0 ≡(p+q) ⋅(pq ′ )

Textbook – Example 1.13

pqd …1

pq d0

d1

+pq

d0

d1

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Set Theory

a ∈A a belongs to Aa ∉A a does not belong to A

A⊂ B if a belongs to A, then a belongs to B (subset)

A=B A⊂ B and A ⊃ B

Empty Set: ∅ Universal Set: U

A⊆ B A ⊂ B and A ≠B

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Set Operations

C =A I B c ∈ C, iff c∈A and c∈B

AI B = {x : x∈A and x∈B}

Aii=1

k

I ≡A1I A2 I K I Ak ={x : x∈Ai , for i =1K k}

Intersection

C =AUB c ∈ C, iff c∈A or c∈B

AU B = {x : x∈A or x ∈B}

Aii=1

k

U ≡A1U A2UK U Ak

={x : if there is an i =1K k, such that x∈Ai}

Union

001 011 010

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More Operations

Set Difference: A−B = {x : x∈A,x∉B}

Symmetric Difference AΔB =(A−B)U(B−A)

Complement ′ A =U −A

Power Set: The set of all subsets of A : P(A)

Cartesian Product A×B : Set of all ordered pairs (a,b)

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Venn Diagrams