HW4_Sol

# HW4_Sol - Homework#4 Solutions 1 Draw a Venn diagram...

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Homework #4 - Solutions 1. Draw a Venn diagram illustrating the p Θ q operation. Solution: p Θ q = pq’ + p’q 2. Write the following in terms of (XOR) and (AND) only: a. p + q Solution: Expand the following diagram (only XOR and AND) to obtain the expression for p + q: p q p’q p q pq’ p q pq’ + p’q

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The above diagram can be expressed as, [p Θ (pq) ] Θ q ------------ 1 It can be shown that the expression 1, can be simplified to p+q: = [p’ (pq) + p (pq)’] Θ q = [0 + p(p’+q’)] Θ q = [pq’] Θ q = (pq’)’q + (pq’)q’ = (p’+q)q + pq’ = p’q + q + pq’ = q(p’+1)+pq’ = q + pq’ = q + p b. (p+q) (p’+q’) = pp’ + pq’ + p’q + qq’ = pq’ + p’q = p Θ q p q
3. A set of gates is said to be functionally complete (or universal) if and only if every Boolean function can be realized entirely by means of gates from this set. a. Prove that the f-gate defined as f (A, B, C) = AB’C+BC’+A’C’ is a universal gate. (Hint: Build NOT, AND, OR gates using only the f-gates) Solution: NOT gate: Consider when c=b = a, f = cc’c + cc’ + c’c’ = 0 + 0 + c’ = c’ Which shows that not gate can be implemented using the given function f. Not gate can be represented as follows:

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HW4_Sol - Homework#4 Solutions 1 Draw a Venn diagram...

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