1. **Problem Statement:** We are given four relations and need to determine which one represents a function. 2. **Definition of a Function:** A relation is a function if every input (x-value) corresponds to exactly one output (y-value). In other words, no x-value repeats with different y-values. 3. **Check each relation:** - (أ) $\{ (1,0), (1,3), (1,4), (0,1) \}$: The input $1$ maps to multiple outputs $0, 3, 4$. This violates the function rule. - (ب) $\{ (1,1), (2,2), (3,2), (4,0) \}$: Each input $1, 2, 3, 4$ maps to exactly one output. This satisfies the function rule. - (ج) $\{ (0,1), (0,12), (0,20), (0,40) \}$: The input $0$ maps to multiple outputs $1, 12, 20, 40$. This violates the function rule. - (د) $\{ (0,1), (10,20), (15,30), (20,10) \}$: Each input $0, 10, 15, 20$ maps to exactly one output. This satisfies the function rule. 4. **Conclusion:** Relations (ب) and (د) represent functions because each input has a unique output. **Final answer:** (ب) and (د) are functions.