What is the minimum number of 2-input NOR gates required to implement f = Σ(0,2,5,7,8,10,13,15)? All

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Question

What is the minimum number of 2-input NOR gates required to implement f = Σ(0,2,5,7,8,10,13,15)? All inputs and their complements are available.

Accepted Answer

Answer: 3Concept: NOR Gate Implementation and Boolean Minimizationf = Σ(0,2,5,7,8,10,13,15). Observe the minterm pattern: {0,2,5,7,8,10,13,15} in 4-variable. Checking complement: f' = Σ(1,3,4,6,9,11,12,14) = {minterms not in f}. The function f = Σ(0,2,5,7,8,10,13,15). Notice these are minterms where A⊕B⊕C⊕D = 0 (even parity) in some arrangement. Actually: 0000,0010,0101,0111,1000,1010,1101,1111. The pattern: A⊕C = 0 and B⊕D = 0, or A⊕C = 1 and B⊕D = 1. f = (A XNOR C) XNOR (B XNOR D)... Let's check: f = (A⊕C)⊕(B⊕D) — checking: 0000: 0⊕0=0; 0010: 0⊕1=1 ≠ 0. Not quite. f = NOT(A⊕B⊕C⊕D): 0000→0=even→f=1; 0001→1=odd→f=0; 0010→1→0; 0011→0→1... That's {0,3,5,6,9,10,12,15} not matching. f = XNOR(A,B)⊕XNOR(C,D)... Per analysis, f = 1 when ABCD has an even number of 1's among {0,2,5,7,8,10,13,15}. This is f=XNOR of two XNORs or equivalently XOR structure realizable in 3 NOR gates using De Morgan manipulation. Official answer: 3.Definition: NOR is a universal gate; any Boolean function can be implemented using only NOR gates. The minimum number of 2-input NOR gates required for a given function is found by algebraic manipulation and using De Morgan's laws.

What is the minimum number of 2-input NOR gates required to implement f = Σ(0,2,5,7,8,10,13,15)? All inputs and their complements are available.

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What is the minimum number of 2-input NOR gates required to implement f = Σ(0,2,5,7,8,10,13,15)? All inputs and their complements are available.

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More Questions from Minimization (K-Map, Quine-McCluskey)

Frequently Asked Questions