Q.Let $A = \{1, 2, 3, 4\}$ and $B = \{2, 4, 6, 8, 10\}$. A relation $R$ from $A$ to $B$ is defined by the rule $y = 2x$, giving $R = \{(1,2), (2,4), (3,6), (4,8)\}$. Examine whether $R$ is a function from $A$ to $B$. If it is, write its domain and range.

A function requires every element of A to have exactly one image in B — check each element of A in turn.

Checking the two conditions for a function.

Every element of $A = \{1,2,3,4\}$ must have an image in $B$, and each must have exactly one image.

From $R = \{(1,2), (2,4), (3,6), (4,8)\}$:

Every element of $A$ (1, 2, 3, 4) appears exactly once as a first co-ordinate, each with a single, unique image. Both conditions for a function are satisfied, so $R$ is a function from $A$ to $B$.

Domain = set of first elements actually used = $\{1, 2, 3, 4\}$ (all of $A$, as required for a function).

Range = set of second elements actually used = $\{2, 4, 6, 8\}$.

Independent cross-check: $10 \in B$ is never used as an image — $10$ belongs to the co-domain $B$ but not the range, exactly why range and co-domain can differ; the rule $y=2x$ applied to each of $1,2,3,4$ indeed reproduces $2,4,6,8$ and never $10$, confirming the range found above.

A common error is confusing the range with the co-domain — writing Range = B = {2,4,6,8,10} instead of only the values actually used, {2,4,6,8}. Another is checking only that no element repeats as a first co-ordinate, while forgetting to check that every element of A is covered.