Project ForgeDigital Logic
Boolean Algebra & Minimization
GATE 2019
Medium
Which one of the following is NOT a valid identity?
- A(x⊕y)⊕z = x⊕(y⊕z)
- (x+y)⊕z = x⊕(y+z)
- Cx⊕y = x+y, if xy=0
- Dx⊕y = (xy+x'y')'
Explanation
Testing option B with x=1,y=1,z=0: LHS=(1+1)⊕0=1⊕0=1. RHS=1⊕(1+0)=1⊕1=0. LHS≠RHS, so this identity is INVALID. (A is XOR's associativity, always true; C is a known special case since xy=0 means x,y are never both 1, making XOR and OR coincide; D is the standard XOR-via-NOR-of-AND/AND-of-complements identity, always true.)
Source: GATE CS 2019 (Set-2) (2019)
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