Prove that for any sets C,D⊆Y and for any function f:X⟶Y, f−1(C∩D)=f−1(C)∩f−1(D).
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Prove that for any sets C,D⊆Y and for any function f:X⟶Y, f−1(C∩D)=f−1(C)∩f−1(D).
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- Let f:AA, where A is nonempty. Prove that f a has right inverse if and only if f(f1(T))=T for every subset T of A.6. For the given subsets and of Z, let and determine whether is onto and whether it is one-to-one. Justify all negative answers. a. b.27. Let , where and are nonempty. Prove that has the property that for every subset of if and only if is one-to-one. (Compare with Exercise 15 b.). 15. b. For the mapping , show that if , then .
- Label each of the following statements as either true or false. Let f:AB where A and B are nonempty. Then f1(f(T))=T for every subset T of B.For the given f:ZZ, decide whether f is onto and whether it is one-to-one. Prove that your decisions are correct. a. f(x)={ x2ifxiseven0ifxisodd b. f(x)={ 0ifxiseven2xifxisodd c. f(x)={ 2x+1ifxisevenx+12ifxisodd d. f(x)={ x2ifxisevenx32ifxisodd e. f(x)={ 3xifxiseven2xifxisodd f. f(x)={ 2x1ifxiseven2xifxisoddLabel each of the following statements as either true or false. 3. Let where A and B are nonempty. Then for every subset S of A.