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IB Math Studies Topic 3 Day 1 – Introduction to Sets Name ________________________ Date ________________ Block ____ A set is ____________________________________________________________________________. SET NOTATION Sets are denoted with a capital letter. The items, names, objects or numbers are enclosed by { } . is the symbol denoting ‘is an element of’ ex. is the symbol denoting “is not an element of’ A = {all even numbers}, therefore 3 A The “bar” symbol “|” is read “such that” ex. F = {x| x is odd} would be read “F is the set of all x, such that x is odd.” The empty set { }, also called the null set, is a set with no elements and is written as SETS OF NUMBERS n = {0, 1, 2, 3, 4, ….} CARDNIAL (NATURAL) NUMBERS Z = {…-3, -2, -1, 0, 1, 2, 3, ….} INTEGERS Q = {m | m = where a and b Z and b 0} RATIONAL NUMBERS R = {all numbers on the real number line} REAL NUMBERS The ________________ ________________ of a set A, n(A), represents the number of ________________ in set A. A finite set has a __________________ number of elements and an infinite set has an infinite number of elements. The _____________ _________ is denoted as { } or . SUMMARY OF TYPES OF SETS TYPE OF SET Null Set (Empty Set) Finite Set Infinite Set Definition A set that has no elements It is denoted by { } or A set whose elements can be counted. A set for which it is impossible to list the elements. Example {x|x2 + 1 = 0, x R } The set {1, 2, 4, 5, 10, 22} contains 6 elements The set {y | y is a real number} cannot be counted, as we cannot write a list of all its elements. THE ALGEBRA OF SETS Set A is said to be equal to set B if both sets are identical, that is, they contain the same elements. We write this as A = B. Set A is said to be equivalent to set B if n(A) = n(B). We write this as A B. ex. Set A = {10, 20, 30, 40, 50} and set B = {a, b, c, d, e}. Set A is equivalent to set B. They are said to have a one-to-one correspondence. (*NOTE: Equivalent means that they have the same number of elements). 10 a 20 b 30 c 40 d 50 e Subsets: Set B is a subset of set A if all elements in set B are contained in set A. Every set is a subset of itself. ex. the set of integers is a subset of the set of real numbers because every integer is also a real number denotes ‘is a subset of’ denotes ‘is not a subset of’ Proper Subsets: Set B is a proper subset of set A if all the elements of B are contained in set A, but the two sets are not the same. A set IS NOT a proper subset of itself. denotes ‘is a proper subset of’ denotes ‘is not a proper subset of’ ex. Let A = {green, yellow, blue} The following are subsets of A: { }, {green}, {yellow}, {blue}, {green, yellow}, {green, blue}, {yellow, blue}, {green, yellow, blue} There are 8 subsets (including the empty set) The following are proper subsets of A: { }, {green}, {yellow}, {blue}, {green, yellow}, {green, blue}, {yellow, blue} There are 7 proper subsets (the set A itself cannot be a proper subset) CORRECT SYMBOL USAGE IN SETS Symbol Correct Usage 2 {1, 2, 4} {2} {1, 2, 4} Incorrect Usage {2} {1, 2, 4} 2 {1, 2, 4} Number of Subsets and Proper Subsets: If n(A) = p, then there are 2P subsets of A and 2P - 1 proper subsets of A. Intersection: The intersection of two sets is the set of elements that they have in ____________ and is denoted by the symbol . ex. A = {even numbers} B = {prime numbers} A B = {2} Union: The union of two sets is the combination of all elements in ____________ sets (each element only being listed once) and is denoted by the symbol . ex. A = {the letters in the word ‘mathematics’} B = {a, b, c, d, e} A B = {a, b, c, d, e, h, i, m, s, t} Number of elements in a union of two sets: n(A B) = n(A) + n(B) - n(A B) **THIS IS AN IMPORTANT CONCEPT IN SET THEORY! ** Complements of Sets and the Universal Set: The universal set is denoted by U. The complement of set A is all of the elements that are in the universal set but are not also in set A. The complement of A is denoted by A’ ex. U = {the letters in the word ‘universal’} A = {a, e, i} A’ = {l, n, r, s, u, v} A A’ = U A A’ = the union of a set and its complement is the universal set the intersection of a set and its complement is the empty set. n(A) + n(A’) = n(U) The sum of the number of elements in a set and its complement equals the total number of elements in the universal set. VENN DIAGRAMS Disjoint sets – set are disjoint if they have no elements in common Intersecting Sets – sets with elements in common U Difference – A\B the set of all element in A and not in B U B A Subsets – a set within a set A U B B A B = A B A U A B A B = B A\B Examples: Shade the area of the Venn diagram noted by the set notation below the universal set. 1. INTERSECTION Disjoint sets Intersecting sets U U A B A A B 2. Subsets B U A B A B A B UNION Disjoint sets Intersecting sets B A A B Subsets U U A B A B U A B A B 3. COMPLEMENT Disjoint sets Intersecting sets Subsets U U B A A A’ 4. B A’ COMPLEMENTS and INTERSECTION Intersecting sets Subsets U U B A A A’ B’ ex A B A’ Disjoint sets U U B A B A’ B A B’ APPLICATIONS A. The classes offered at a summer camp are Crafts(C), Volleyball (V) and Swimming (S). The Venn diagram shows the numbers of children involved in each activity. U C V 8 1 5 3 2 S (a) (b) (c) (d) 4 6 4 How many children are in only the Crafts class? How many children are in both the Crafts and Volleyball class? How many children are not in the Crafts class? Shade the part of the Venn diagram that represents the set C’ S. ____________ __________ __________ What does this represent? ex B: During a two week period, Murielle took her umbrella with her on 8 days, it rained on 9 days, and Murielle took her umbrella on five of the days when it rained. (a) Display this information on a Venn diagram: (b) Find the number of days that Murielle did not take her umbrella and it rained. (c) Find the number of days that Murielle did not take her umbrella and it did not rain.