EXERCISES 3: LECTURE FOUNDATIONS OF MATHEMATICS
Exercise 1. LetX, Y, Z be sets. Moreover, let f:X→Y andg:Y →Z be maps. Show:
(a) Iff and g are injective, theng◦f injective.
(b) Iff and g are surjective, theng◦f surjective.
(c) f is injective if and only if there exists h:Y →X such thath◦f = idX. (d) f is surjective if and only if there existsh:Y →X such thatf ◦h= idY. Above idX resp. idY denote the identity maps onX resp. Y.
Exercise 2. LetX, Y be sets. Further, letf:X →Y be a map whose preimage is denoted byf−1. Show that the following are equivalent:
(i) f is injective.
(ii) f−1(f(A)) =A for allA⊂X.
(iii) f(A∩B) =f(A)∩f(B) for allA, B ⊂X.
(iv) For allA, B⊂X withA∩B =∅ one has f(A)∩f(B) =∅.
(v) For all A, B⊂X withB⊂Aone has f(A\B) =f(A)\f(B).
Exercise 3. Let W, X, Y, Z be sets, and f: W → X, g:X → Y and h: Y → Z be maps.
Show thatf, g, hare bijective in case g◦f andh◦g are.
Exercise 4. Let X, Y be sets, and letf:X → Y be a map whose preimage is denoted by f−1. Let A, B be subsets ofX andC, D be subsets ofY.
Decide which of the following statements are true and which are false.
(a) IfA6=∅, thenf(A)6=∅.
(b) IfC 6=∅, thenf−1(C)6=∅.
(c) IfA⊂B, then f(A)⊂f(B).
(d) IfC ⊂D, thenf−1(C)⊂f−1(D).
(e) f(A∩B) =f(A)∩f(B).
(f) f−1(C∩D) =f−1(C)∩f−1(D).
(g) f(A∪B) =f(A)∪f(B).
(h) f−1(C∪D) =f−1(C)∪f−1(D).
(i) If B ⊂A, then f(A\B) =f(A)\f(B).
(j) IfD⊂C, thenf−1(C\D) =f−1(C)\f−1(D).
Justify your answer with a proof or a counterexample.
Submission of the exercise sheet: 15.Oct.2018 before the lecture. Return of the exercise sheet: 18.Oct.2018 during the exercise classes.
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