By Baker.

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Proof. Let f : m −→ X and g : n −→ X be bijections. Using the inverse g −1 : X −→ n which is also a bijection, we can form a bijection h = g −1 ◦ f : m −→ n. By part (c), m = n. For a finite set X, the unique n ∈ N0 for which there is a bijection n −→ X is called the cardinality of X, denoted |X|. If X is infinite then we sometimes write |X| = ∞, while if X is finite we write |X| < ∞. 3 without proof to give some important facts about cardinalities of finite sets. 5 (Pigeonhole Principle). Let X, Y be two finite sets.

15. Let T ⊆ R2 be an equilateral triangle B A1 1 111 11 11 11 · O 111 11 1 with vertices A, B, C. C Then a symmetry is defined once we know where the vertices go, hence there are as many symmetries as permutations of the set {A, B, C}. Each symmetry can be described using permutation notation and we obtain the 6 symmetries ι= A B C , A B C A B C , B C A A B C , C A B A B C , A C B A B C C B A A B C . B A C Therefore we have | Sym( )| = 6. 16. Let S ⊆ R2 be the square B(−1, 1), C(−1, −1), D(1, −1).

Each f (xk ) can be chosen in n ways so the total number of choices is nm . Hence |Y X | = nm . , Y = {0, 1}. The set {0, 1}X is called the power set of X, and has 2|X| elements and indeed it is often denoted 2|X| . It has another important interpretation. For any set X, we can consider the set of all its subsets P(X) = {U : U ⊆ X is a subset}. Before stating and proving our next result we introduce the characteristic or indicator function of a subset U ⊆ X, χU : X −→ {0, 1}; χU (x) = 1 if x ∈ U , 0 if x ∈ / U.

### Algebra and number theory, U Glasgow notes by Baker.

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