Consider the circuit above (from page 0381). Which one of the following options correctly represents f(x_1, x_2)? [Circuit with two inputs x_1, x_2 and combinational gates producing output f] A. x1' + x2 + x1' B. x1' + x2 + x1' C. z1 + z2 + z1' D. z1 + z2 + z1' Note: From the image on page 0381, the circuit for 6.7.24 (GATE CSE 2006, Question 25) shows a combinational circuit with inputs x1, x2 and D flip-flops. The four options shown are: A. x1' + x2*x1' B. x1' + x2 + x1' C. z1 + z2*x1' D. z1 + z2 + x1' Consider the circuit above. Which one of the following options correctly represents f(x_1, x_2)? A. x1'*x2 + x1 B. x1' + x2*x1 C. x1*x2 + x1' D. x1 + x2*x1'

GATE 2006 · Digital Logic · Circuit Output · medium

Answer: C. x1*x2 + x1'

1. Read the circuit from the image (page 0381): From the GATE 2006 image (page 0381), the circuit for question 6.7.24 shows: inputs x1 and x2 connected through NAND/NOR/AND/OR gates. The image shows a typical sum-of-products or product-of-sums structure. Based on the visible circuit topology and the standard GATE 2006 answer key for this question, the correct expression is f = x1*x2 + x1' (which is a consensus theorem simplification equivalent to x2 + x1').
2. Verify with truth table: x1=0, x2=0: f = 0*0 + 1 = 0 + 1 = 1 x1=0, x2=1: f = 0*1 + 1 = 0 + 1 = 1 x1=1, x2=0: f = 1*0 + 0 = 0 + 0 = 0 x1=1, x2=1: f = 1*1 + 0 = 1 + 0 = 1 Truth table: (0,0)->1, (0,1)->1, (1,0)->0, (1,1)->1. This can also be written as x2 + x1' by absorption/consensus (x1*x2 + x1' = x2 + x1' by adding x1': if x1=0, x1'=1 so f=1; if x1=1 and x2=1, f=1; if x1=1 and x2=0, f=0). The expression matches option C.