The relation (R={(x,y):|x-2y|=2,\ x\in{0,2,4},\ y\in{-1,0,1,2,3}}) is considered from the given domain to codomain. Why is it not a function?

Correct answer: Because (x=2) has two images

The direct answer is A. A relation is a function only when every domain element has exactly one image. For x=2, substitute into |x−2y|=2: |2−2y|=2. This means 2−2y=2 or 2−2y=−2. The first equation gives y=0, and the second gives y=2. Thus the same input x=2 has two images, 0 and 2, so the relation is not a function. Option A is correct for precisely this reason. Option B is wrong because x=0 does have images: |0−2y|=2 gives y=1 or y=−1, both in the codomain. Option C is wrong because x=4 also has images: |4−2y|=2 gives y=1 or y=3. Option D is irrelevant and wrong; the mere presence of y=3 in the codomain does not violate the function rule. The problem is two outputs for one input, not the size of the codomain. Memory cue: a function allows many inputs to share one output, but never one input to have two outputs.