Q.Determine which of the following relations are functions from $A$ to $B$. Give reasons.

Step 1: For (i) $\{(1,2),(1,3),(2,3),(4,5)\}$: the first component $1$ appears twice, once paired with $2$ and once with $3$ — so $f(1)$ would not be a single well-defined value.

Step 2: Hence (i) is a relation but NOT a function.

Step 3: For (ii) $\{(1,4),(2,4),(3,4),(5,4)\}$: the first components $1,2,3,5$ are all distinct, each appearing exactly once, even though they share the same image $4$.

Step 4: A repeated image (many inputs to one output) is allowed for a function; only a repeated first component is not. So (ii) IS a function (a constant one).

Check each relation's list of first components for repeats; a repeat with different second components disqualifies it as a function, but a repeated second component (image) alone does not.

• Rejecting (ii) by mistakenly thinking a repeated output value is not allowed.
• Accepting (i) by only checking that all four pairs are distinct, without checking whether any first component repeats.