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Unformatted text preview: Definitions and Theorems Chapter 0 1. Cartesian Product 2. Relation, domain, image 3. Function 4. Composition of two functions 5. injective (11), surjective (onto), bijective, inverse function 6. Principle of Mathematical Induction (p 12) 7. Equivalence relation: A relation R on a set A is said to be an equivalence relation if: (a) (reflexivity) For each x ∈ A we have ( x,x ) ∈ R (b) (symmetry) If ( x,y ) ∈ R then ( y,x ) ∈ R (c) (transitivity) If ( x,y ) ∈ R and ( y,z ) ∈ R , then ( x,z ) ∈ R If ( x,y ) ∈ R then we write x ∼ y 8. Let U be a family of sets that includes at least R and all of its subsets. On this set we define an equivalence relation ∼ as follows: A ∼ B if there exists a bijection f : A → B . If A ∼ B then we say A and B are (cardinally) equivalent . 9. Let S be a set. Its power set P is the family of all subsets of S . Sometimes the power set of S is denoted by 2 S . Hence P ≡ 2 S := { A  A ⊂ S } ....
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 Spring '07
 thieme
 Mathematical Induction, Empty set, Order theory, Natural number, Georg Cantor

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