Given:
f ⊆ Z × Z defined by
f = { (ab, a + b) : a, b ∈ Z }
Step 1: Recall definition of a function
A relation f from Z to Z is a function if
for every element in the first component, there is one and only one corresponding element in the second component.
Step 2: Check uniqueness of image
Here, the first component is ab and the second component is a + b.
Consider the element 0 in Z.
If a = 0, b = 2:
ab = 0, a + b = 2
So, (0, 2) ∈ f
If a = 1, b = 0:
ab = 0, a + b = 1
So, (0, 1) ∈ f
Thus, the same first element 0 has two different images, 1 and 2.
Step 3: Conclusion
Since one element of Z has more than one image in Z,
the relation f does not satisfy the definition of a function.
Final Answer:
f is not a function from Z to Z,
because the same first element can correspond to different second elements.