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MAT371defs0

# MAT371defs0 - Definitions and Theorems Chapter 0 1...

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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 (1-1), 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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MAT371defs0 - Definitions and Theorems Chapter 0 1...

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