Question

6. Let A = {1, 2, 3, 4} and B = {5, 6, 7}. Let f = {(1, 5),(2, 5),(3, 6),(x, y)} where x ∈ A and y ∈ B are to be determined by you. (a) In how many ways can you pick x ∈ A and y ∈ B such that f is not a function? (b) In how many ways can you pick x ∈ A and y ∈ B such that f : A → B is onto? (c) In how many ways can you pick x ∈ A and y ∈ B such that f : A → B is not

Answer 1

(a). For f not to be a function, we have to ensure that f has more than one image for some point in A. Now, the images of 1,2 and 3 are defined. So if we take x=4 then whatever y is, f will always be a function. So we can't take x=4. We have three choices for x, namely 1 ,2 and 3. If we take x=1, then we can't take y=5 as then f will be a function. So we have to take y as 6 or 7. Similarly for x=2 we have to take y=6 or 7 and for x=3, we have take y=5 or 7. So, in 6 ways x and y can be picked such that f is not a function. [ Namely these six ways are (1,6),(1,7),(2,6),(2,7),(3,5),(3,7). ]

(b). f is already defined on 1,2 and 3. So we have to take x=4. Now, image of f is {5,6}. So, 7 is not there and hence for f to be onto we must take y=7. So, (4,7) is the only choice for (x,y) such that f will be onto.

(c). Again we have to take x=4. Now if we take y=7, then f will be onto. So, we can't take y=7. Hence, we can take y=5 or 6. So here x and y can be picked in 2 ways such that f will not be onto. The choices are (4,5),(4,6).