What is the Boolean expression for F(w,x,y,z) in GATE CS 2025 Set-2 Q42?

Option D is correct: F = x̄z̄ + xz. The K-Map shows that the function depends only on variables x and z — specifically, F is 1 whenever x and z have the same value (both 0 or both 1). This is the XNOR of x and z, expressed as x̄z̄ + xz.

How do you read a 4-variable K-Map for F(w,x,y,z)?

A 4-variable K-Map is a 4×4 grid. Group adjacent 1s in powers of 2 (1, 2, 4, 8, 16). Larger groups give simpler terms. If the 1s form groups that span all values of w and y, those variables drop out of the expression, leaving only the relevant variables in the simplified Boolean expression.

Why do options A, B, and C not correctly represent F in GATE CS 2025 Q42?

Options A, B, and C include unnecessary minterms or incorrect terms that don't match the K-Map. They either cover cells that are 0 in the K-Map or miss valid groupings that simplify to the cleaner expression x̄z̄ + xz shown in Option D.

What is the XNOR gate connection to K-Map F(w,x,y,z) in GATE CS 2025 Q42?

The expression x̄z̄ + xz is the XNOR of x and z, written as x XNOR z. It evaluates to 1 when both inputs are equal (both 0 or both 1) and 0 when they differ. The K-Map pattern confirms F depends only on x and z, with w and y being don't care variables that cancel out during grouping.

K-Map Boolean Function F(w,x,y,z) | GATE CS 2025 Set-2 Q42 Digital Logic

Computer Sciences > GATE 2025 SET-2 > Karnaugh Maps

Given the following Karnaugh Map for a Boolean function F(w,x,y,z):

Which one or more of the following Boolean expression(s) represent(s) F?

w̅x̅y̅z̅+wx̅y̅z̅+w̅x̅yz̅+wx̅yz̅+xz

w̅x̅y̅z̅+w̅x̅yz̅+wx̅yz+xz

w̅x̅y̅z̅+wx̅y̅z̅+wx̅y̅z+xz

x̅z̅+xz

Correct : d

The correct answer is Option D - x̄z̄ + xz.
From the given K-Map for F(w, x, y, z), the key observation is that the function''s value depends only on variables x and z - the variables w and y don''t affect the output at all. This becomes clear when you group the 1s on the K-Map: the groups span all possible values of w and y, causing those variables to cancel out completely.
What remains is a function that is 1 whenever x and z have the same value - both 0 or both 1. This is exactly the XNOR of x and z:
F = x̄z̄ + xz (x XNOR z)
Option A, B, C all include extra minterms or incorrect terms involving w and y that don''t match the K-Map. They either cover 0-cells or fail to simplify into the cleanest groupings. All three are incorrect.
Option D is the fully minimized and correct expression - just two 2-literal terms, confirming the K-Map reduces entirely to an XNOR relationship between x and z.

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