Sets and Functions. MATH 464/506, Real Analysis. J. Robert Buchanan. Summer Department of Mathematics. J. Robert Buchanan Sets and Functions

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1 Sets and Functions MATH 464/506, Real Analysis J. Robert Buchanan Department of Mathematics Summer 2007

2 Notation x A means that element x is a member of set A. x / A means that x is not a member of A.

3 Subsets A B or B A means that every element of set A is also a member of set B. We say that A is a subset of B. A B means that every element of A is also a member of B and there is at least one element of B that is not a member of A. We say A is a proper subset of B.

4 Set Equality Two sets A and B are said to be equal, denoted as A = B, if they contain the same elements. Remark: often to prove A = B we will show that A B and B A.

5 Common Sets Natural numbers N = {1, 2, 3,...} Integers Z = {0, 1, 1, 2, 2,...} Rational numbers Q = { m n : m, n Z and n 0} Real numbers R

6 Common Sets Natural numbers N = {1, 2, 3,...} Integers Z = {0, 1, 1, 2, 2,...} Rational numbers Q = { m n : m, n Z and n 0} Real numbers R Other examples

7 Set Operations The union of sets A and B is the set A B = {x : x A or x B}. The intersection of sets A and B is the set A B = {x : x A and x B}. The complement of B relative to A is the set A\B = {x : x A and x / B}.

8 Set Operations The union of sets A and B is the set A B = {x : x A or x B}. The intersection of sets A and B is the set A B = {x : x A and x B}. The complement of B relative to A is the set A\B = {x : x A and x / B}. Diagrams

9 DeMorgan Laws A set with no elements is called the empty set, denoted by. Two sets A and B are disjoint if A B =.

10 DeMorgan Laws A set with no elements is called the empty set, denoted by. Two sets A and B are disjoint if A B =. Theorem If A, B, and C are sets, then 1 A\(B C) = (A\B) (A\C), 2 A\(B C) = (A\B) (A\C),

11 Collections of Sets Consider the finite collection of sets {A 1, A 2,..., A n }. The union of this collection is the set of elements which appear in at least one of the sets A k. The intersection of this collection is the set of elements which appear in all of the sets A k.

12 Collections of Sets Consider the finite collection of sets {A 1, A 2,..., A n }. The union of this collection is the set of elements which appear in at least one of the sets A k. The intersection of this collection is the set of elements which appear in all of the sets A k. Consider the infinite collection of sets {A 1, A 2,..., A n,...}. Again we may consider the union and intersection of these sets. A n = {x : x A n for some n N} n=1 A n = {x : x A n for all n N} n=1

13 Cartesian Products and Functions If A and B are nonempty sets, the Cartesian product A B of A and B is the set of all ordered pairs (a, b) with a A and b B. A B = {(a, b) : a A and b B}

14 Cartesian Products and Functions If A and B are nonempty sets, the Cartesian product A B of A and B is the set of all ordered pairs (a, b) with a A and b B. A B = {(a, b) : a A and b B} Let A and B be sets. A function from A to B is a set f of ordered pairs in A B such that for each a A there exists a unique b B with (a, b) f.

15 Domain and Range The set of first elements of function f is called the domain of f and can be denoted D(f). The set of all second elements of function f is called the range of f and can be denoted R(f). D(f) = A but R(f) B

16 Vertical Line Test The essential condition of a function f : A B is that if (a, b) f and (a, b ) f then b = b.

17 Vertical Line Test The essential condition of a function f : A B is that if (a, b) f and (a, b ) f then b = b. Notation: If (a, b) f then we can write f(a) = b and we say that b is the value of f(a) or that b is the image of a under f.

18 Direct and Inverse Images If E is a subset of A, then the direct image of E under f is the subset f(e) of B given by f(e) = {f(x) : x E}. If H is a subset of B, then the inverse image of H under f is the subset f 1 (H) of A given by f 1 (H) = {x A : f(x) H}.

19 Direct and Inverse Images If E is a subset of A, then the direct image of E under f is the subset f(e) of B given by f(e) = {f(x) : x E}. If H is a subset of B, then the inverse image of H under f is the subset f 1 (H) of A given by f 1 (H) = {x A : f(x) H}. Examples

20 Special Types of Functions Let f : A B be a function from A to B. 1 The function f is said to be injective (or to be one-to-one) if whenever a 1 a 2, then f(a 1 ) f(a 2 ). If f is an injective function, we may also call f an injection. 2 The function f is said to be surjective (or to map A onto B) if f(a) = B, i.e. R(f) = B. If f is a surjective function, we may also call f a surjection. 3 If f is both injective and surjective, then f is said to be bijective. If f is bijective, we may also call f a bijection.

21 Special Types of Functions Let f : A B be a function from A to B. 1 The function f is said to be injective (or to be one-to-one) if whenever a 1 a 2, then f(a 1 ) f(a 2 ). If f is an injective function, we may also call f an injection. 2 The function f is said to be surjective (or to map A onto B) if f(a) = B, i.e. R(f) = B. If f is a surjective function, we may also call f a surjection. 3 If f is both injective and surjective, then f is said to be bijective. If f is bijective, we may also call f a bijection. Example

22 Inverse Functions If f : A B is a bijection of A onto B, then g = {(b, a) B A : (a, b) f } is a function on B into A. This function is called the inverse function of f, and is denoted by f 1.

23 Inverse Functions If f : A B is a bijection of A onto B, then g = {(b, a) B A : (a, b) f } is a function on B into A. This function is called the inverse function of f, and is denoted by f 1. Example

24 Composition of Functions If f : A B and g : B C, and if R(f) D(g) = B then the composite function g f is the function from A into C defined by (g f)(x) = g(f(x)) for all x A.

25 Composition of Functions If f : A B and g : B C, and if R(f) D(g) = B then the composite function g f is the function from A into C defined by (g f)(x) = g(f(x)) for all x A. Example

26 Composition of Functions If f : A B and g : B C, and if R(f) D(g) = B then the composite function g f is the function from A into C defined by (g f)(x) = g(f(x)) for all x A. Example Theorem Let f : A B and g : B C be functions and let H be a subset of C. Then we have (g f) 1 (H) = f 1 (g 1 (H))

27 Restrictions of Functions If f : A B is a function and if A 1 A we can define the function f 1 : A 1 B by f 1 (x) = f(x) for x A 1. The function f 1 is called the restriction of f to A 1 and may be denoted f A 1.

28 Restrictions of Functions If f : A B is a function and if A 1 A we can define the function f 1 : A 1 B by f 1 (x) = f(x) for x A 1. The function f 1 is called the restriction of f to A 1 and may be denoted f A 1. Example

29 Homework Read Sections 1.1 and 1.2. Pages 11-12, exercises 3, 4, 7, 8, 13, 14, 15, 19 Pages 15-16, exercises 1 5, Numbered exercises in boxes are to be written up separately and submitted for grading at class time on Friday.