1. Let f:A->B, where A and B are nonempty, and let T1 and T2 be subsets of B.
a. Prove that f-1 (T1 U T2) = f-1(T1) U f-1(T2)
b. Prove that f -1 (T1 n T2) = f ^-1 (T1) n f ^-1(T2)
c. Prove that f ^-1(T1) – f ^-1(T2) = f ^-1(T1 -T2)
d. Prove that if T1 is a proper subset of T2, then f ^-1(T1) is a proper subset of f ^-1(T2)
2. Let f:A?B, where A and B are nonempty
a. Prove that f(S1) – f(S2) is a proper subset of f(S1 – S2) for all subsets S1 and S2 of A.
b. Give an example where there are subsets S1 and S3 of A such that
f(S1) – f(S2) ? f(S1 – S2)
3. Let A be a set of integers closed under subtraction
a. Prove that if A is nonempty, then 0 is in A.
b. Prove that if x is in A then -x is in A.
4. Prove that the cancellation law for multiplication holds in Z. That is if xy = xz and x ? 0, then
y=z.
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1. Let f:A->B, where A and B are nonempty, and let T1 and T2 be subsets of B. a. Prove that f-1 (T1 U T2) = f-1(T1) U f-1(T2) b. Prove that f -1 (T1 n T2) = f ^-1 (T1) n f ^-1(T2) c. Prove that f ^-1(T1) – f ^-1(T2) = f ^-1(T1 -T2) d. Prove that if T1 is a proper subset of T2, then f ^-1(T1) is a proper subset of f ^-1(T2) 2. Let f:A?B, where A and B are nonempty a. Prove that f(S1) – f(S2) is a proper subset of f(S1 – S2) for all subsets S1 and S2 of A. b. Give an example where there are subsets S1 and S3 of A such that f(S1) – f(S2) ? f(S1 – S2) 3. Let A be a set of integers closed under subtraction a. Prove that if A is nonempty, then 0 is in A. b. Prove that if x is in A then -x is in A. 4. Prove that the cancellation law for multiplication holds in Z. That is if xy = xz and x ? 0, then y=z.