In which option is the relation not a function from A = {0,1,2,3} to B = {p,q,r}, even though every element of A appears as a first component?

Correct answer: {(0,p),(1,q),(2,q),(3,r),(0,r)}

A relation from A to B is a function precisely when each element of A occurs as the first component of exactly one ordered pair. It is acceptable for different inputs to share the same output; this is why options A, B, and D are functions even though some second components repeat. In option C, the input 0 appears twice: it is paired with p and also with r. Thus 0 has two distinct images, while the other inputs have one image each. This violates the uniqueness condition, so option C is the relation that is not a function. The fact that every element of A appears at least once is necessary but not sufficient; it must appear exactly once.