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EXERCISES 4: LECTURE FOUNDATIONS OF MATHEMATICS

Exercise 1. LetX={a, b, c}. List all possible equivalence relations onX.

Exercise 2. Let X be a set, and let XX denote the set of all maps X → X. Further, let S(X) denote the set of all bijective mapsX→X. Show:

(a) Iff, g∈S(X), theng◦f and f◦g are also in S(X).

(b) IfX has at least two elements, thenXX is not commutative with the operation given by◦.

(c) If X has at least three elements, then S(X) is not commutative with the operation given by◦.

Exercise 3. LetX, Y be sets and let∼X,∼Y be equivalence relations on these sets. Moreover, letf:X→Y be a map such that

(?) : (x1X x2)⇒(f(x1)∼Y f(x2)) ∀x1, x2∈X.

Show that there is a unique map[f]such that X f //

pX

Y

pY

X/∼X

[f]

//Y /∼Y

commutes. What happens if(?) in the case where ∼X is the identity relation? (Meaning that (x1X x2)⇔(x1 =x2).)

Exercise 4. Let (X,≤) be an ordered set. Further, let A and B subsets of X which are bounded above. Show the following statements in the case where the corresponding suprema und infima exist:

(a) sup(A∪B) = sup(sup(A),sup(B)).

(b) IfA⊂B, then sup(A)≤sup(B).

(c) IfA∩B6=∅, thensup(A∩B)≤inf(sup(A),sup(B)).

Formulate and prove the corresponding statements for subsets C andD of X which are bounded below.

Submission of the exercise sheet: 22.Oct.2018 before the lecture. Return of the exercise sheet: 25.Oct.2018 during the exercise classes.

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