1 Chapter 1: SETS. 1.1 Describing a set

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1 1 Chapter 1: SETS set is a collection of objects The objects of the set are called elements or members Use capital letters :, B, C, S, X, Y to denote the sets Use lower case letters to denote the elements: a, b, c, x, y If x is an element of the set X, we write x X If x is not an element of the set X, we write x / X 11 Describing a set 1 List all elements if the set consists of a small number of elements: X = {a, b, c} = {1, 2,, 100} need to list the first two elements to if they are consecutive S = {1, 3, 5, } list the first 3 elements to give away the pattern(not correct to list: S = {1, 3, 5, 7, }, which is redundant, nor S = {1, 3, } because of not enough information) NOTE: = {1, 2, 3} = {2, 1, 3} = {1, 1, 3, 2} = {} is the empty set { } as the first set has one element, and the second set has no elements 2 set with condition(s): S = {x p(x)} or {x : p(x)}, that is: S contains all the elements x that satisfy the condition (or have the property) p(x), where p(x) a property that depends on x Ex: = {x : x is even } S = {x : (x 1)(x + 2) = 0} T = {x : x = 1} X = {x : x is a student in M1025 } more complex example: Let = {1, 2,, 10} Then define B = {x : x < 7} = {1, 2, 3, 4, 5, 6} For a set S, we write S to denote the number of elements in the set S The number S is called the cardinality of S Ex: for the examples above: S = 2, {4} = 1, = 0, { } = 1 1

2 Special sets N = {1, 2,, } is the set of all positive whole numbers Z = {, 2, 1, 0, 1, 2, } is the set of integers (whole numbers) E = {y : y is an even integer } = {, 4, 2, 0, 2, 4, } Q = { p q : p, q Z, q 0} is the set of the rational numbers I = the set of irrationals, for example: π, 2, 3 R = the real numbers R + = the positive real numbers C = the set of complex numbers: a + bi 12 Subsets set is a subset of a set B if every element of is an element of B If is a subset of B, we write B N Z Q R C If a set is not a subset of a set B, we write B In this case, there is an element in the set that is not in B The empty set is a subset of every set INTERVLS: Let a, b R [a, b] = {x R : a x b} [a, b) = {x R : a x < b} (a, b] = {x R : a < x b} (a, b) = {x R : a < x < b} (a, ) = {x R : a < x} (, b] = {x R : x b} Two sets and B are equal, written as = B, if they have exactly the same elements By definition = B means B and B If they are not equal then we write B set is a proper subset of a set B if B and B, and it is denoted by B Ex: Let = {a} and B = {a, b} When we talk about subsets, we are concerned with subsets of a larger set, usually called universal set denoted by U We can use Venn Diagrams to represent a set: 2

3 : B: Figure 1: Venn Diagram of two Sets w : x y z B: Figure 2: Venn Diagram From the digram: x, y B, z, z B, w /, w / B For a set, the power set P() of is the set of all subsets of Ex: = {a, b} Then P() = {, {a}, {b}, {a, b}} = {, a, b, {a, b}} since a, b are not sets without the curly brackets P() = 4 = 2 2 = 2 In general: P() = 2 13 Set Operations Let and B be two sets The following are ways of combining two or more sets: 1 The intersection of and B: B = {x : x and x B} If B =, then and B are disjoint 2 The union of and B: B = {x : x or x B} 3 The difference of and B: \ B = {x : x and x / B} 3

4 B B B B \ B B Ā Figure 3: Set operations 4 The complement of : Ā = {x : x / } = U \, where U is the universal set 5 The relative complement of B in (same as the difference of the two sets): \ B = {x : x and x / B} 14 Indexed Collection of Sets Suppose that 1, 2,, n is a collection of collection of sets, (n 3) The following are definitions of ways of combining two or more sets: 1 The intersection of the n sets 1, 2,, n is: n i = {x : x i,for all values i, 1 i n} i=1 2 The union of of the n sets 1, 2,, n is: 4

5 n i = {x : x i, for at least one value i, 1 i n} i=1 Example: Let i = {i, i + 1}, 1 i 10 What are the intersection and the union of them all? 1 10 i=1 i = 2 10 i=1 i = {1, 2,, 11} Note: If we have different index sets, we have different results: Let i = {i, i + 1}, and the index set I = {1, 5, 10} Then 1 i I i = 2 i I i = {1, 2, 5, 6, 10, 11} Let I be the indexed set For each α I there is a set, which is deonoted by S α that corresponds to α Hence {S α } α I is called an indexed collection of sets We then have S α, s α α I α I Example 1: Let n = [ 1 n, 1 n ] = {x : 1 n x 1 n }, and let I = N What is their union and intersection? n I n = [ 1, 1], n I n = {0} Example 2: Let S n = {n, 2n}, and let I = {1, 2, 3, 4} What are S n for n = 1, 2, 3, 4? What is their union and intersection? S 1 = {1, 2}, S 2 = {2, 4}, S 3 = {3, 6}, S 4 = {4, 8} S n = {1, 2, 3, 4, 6, 8}, S n = n I n I 5

6 15 Partitions of Sets Let S be a set of subsets of a set Ex: = {a, b, c, d} and S = {{a}, {b, c}, {d}} Then S is pairwise disjoint set if every two distinct subsets in (that belong to) S are disjoint If S is pairwise disjoint set of nonempty subsets of such that every element of belongs to some subset of S, then S is called a partition of Ex: Let = {1, 2,, 7} Which of the following are partitions? S 0 = {{1}, {1, 2, 3}, {4, 5}, {3, 6, 7}} No S 1 = {{1}, {2, 3}, {4, 5, 6}, {7}} Yes S 2 = {{1}, {1, 2, 3}, {5, 6}} No S 3 = {, {1}, {2, 3, 4, 5, 6}} No 2, 3 6, Figure 4: partition of the set That is: S is a partition if: 1 if X S, then X, 2 if X, Y S, then X Y =, 3 X S X = nother example: S = {{1, 2, 5}, {3, 4}, {6, 7}} 6

7 16 Cartesian Product The Cartesian product of two sets and B is B = {(a, b) : a and b B} Note that (a, b) is an ordered pair, so (a, b) (b, a) (versus the set {a, b} = {b, a}) Example: Let = {x, y} and B = {1, {2}, 3} Then B = {(x, 1), (x, {2}), (x, 3), (y, 1), (y, {2}), (y, 3)} B = {(1, x), (1, y), ({2}, x), ({2}, y), (3, x), (3, y)} Note that B B What is B? B = B = 6 in this case If =, then B = and B =, for any set B, by definition 7

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