A Set A set is a collection of objects. The objects are called members or elements. The elements may have something in common or may be unrelated.

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1 A Set A set is a collection of objects. The objects are called members or elements. The elements may have something in common or may be unrelated. A = {5, 10, 15, 20, 25} B = {prime numbers less than 20} C = {cat, red, 17, geography} The curly brackets {...} are the symbol for the set of.

2 Members of a Set The members or ELEMENTS of a set are the objects that belong to the set. If A = {2, 6, 9} then the elements of A are 2, 6 and 9. We use the symbol to say is an element of. So we have 2 A and 6 A and 9 A. The symbol means is not an element of. So we can say 10 A.

3 The Universal Set The Universal set is the set from which all the other sets are formed. The symbol for Universal set is U If A = {1, 2, 3} and B = {4, 5, 6} then the universal set could be {1, 2, 3, 4, 5, 6}. In fact the Universal set is any set that contains the elements 1, 2, 3, 4, 5 and 6. So {whole numbers less than 10} might also have been the Universal set.

4 The Empty Set Any set which has no elements is called an empty set. There are two symbols for the empty set. They are { } or 1. { 17 year old students in Form 1 } = { } 2. { prime numbers that end in 0} = 3. {3, 5, 7, 9,...}

5 The Union of Two Sets The union of two sets A and B is a set which contains all the elements in A or in B or in both. The elements are written in order and are not repeated. The symbol that we use for union is. 1. If A = { 2, 4, 6, 8, 10 } and B = { 3, 4, 6, 8, 9 } then A B = { 2, 3, 4, 6, 8, 9, 10 } 2. If S = { a, c, d, e, g } and T = { b, c, d, f } then S T = { a, b, c, d, e, f, g }

6 The Intersection of Two Sets The intersection of two sets is a set which contains those elements which belong to both sets. The symbol for intersection is. 1. If C = { 11, 13, 14, 16, 23, 28 } and D = { 12, 13, 15, 16, 21, 23 } then C D = { 13, 16, 23 } 2. If K = {a, b, c, d} and L = {e, f, g, h, i} then K L = { } since the sets have no elements in common.

7 Subsets of a Set(1) When we are given a set, we may make other sets using the elements in the given set. These new sets that we make are called SUBSETS of the given set. If A = {apple, banana, coconut} then we can make subsets of A by using the elements in A. The subsets of A are {apple}, {banana}, {coconut}, {apple, banana}, {apple, coconut}, {banana, coconut}. This list of subsets is not complete. It also contains the sets { } and {apple, banana, coconut}. So there are 8 subsets of A.

8 Subsets of a Set(2) The symbol for is a subset of is {2, 5} {1, 2, 3, 4, 5} {3, 5, 7} {odd numbers} {a, b, c, d, e} {a, b, c, d, e} {a, b, c, d, e} Every set will have the empty set and the set itself as subsets.

9 The Complement of a Set The complement of a set S is the set whose elements belong to the Universal set but NOT to set S. The symbol for the complement of S is S c or S /. 1. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {1, 4, 5, 7, 8, 10} then A c = { 2, 3, 6, 9 } 2. If U = {a, b, c, d, e, f, g, h, i, j, k, l} and M = {b, c, e, g, j, k} then M / = { a, d, h, i, l }

10 Finite and Infinite Sets A set is finite if the list of elements stops. If the list of elements is never-ending, then the set is infinite. 1. P = {factors of 18} P is a finite set. 2. T = {multiples of 7} T is an infinite set. 3. W = {1, 2, 3,..., 67} W is a finite set. 4. X = {1, 4, 9, 16, 25,...} X is an infinite set.

11 Equal and Equivalent Sets Two sets are equal if they have exactly the same elements. Two sets are equivalent if they have the same number of elements. 1. {2, 5, 7} and {7, 2, 5} are equal sets because they both contain the elements 2, 5 and {cat, dog, egg, fish} and {134, p, John, tree} are equivalent sets because they both contain 4 elements.

12 Venn Diagrams(1) Sets may be represented on a diagram known as a Venn Diagram. If U = {1, 2, 3,..., 10}, A = {2, 4, 6, 7, 9} and B = {2, 3, 4, 7, 8, 9} then the Venn diagram is U 5 A 6 B Venn Diagrams (2) 10 1

13 Venn Diagrams (2) If U = {21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32}, X = {23, 24, 26, 27, 29} and Y = {22, 23, 24, 26, 27, 28, 29, 31} then the Venn diagram is 25 Y X Venn Diagrams (2) 30 U

14 Venn Diagrams (3) If U = {a, b, c, d, e, f, g, h, i, j}, P = {a, c, h, j} and Q = {b, d, f} then the Venn diagram is U P c a j g b d Q e h Venn i Diagrams (2) f

15 Problem Solving(1) In a class of students, 12 love Maths and English, 8 love Maths only, 7 love English only and 3 love neither. Draw a Venn Diagram and use it to state the number of children in the class. U = {students}, M = {students who love Maths}, E = {students who love English} U = 30 So the number of students in the class is M E

16 Problem solving (2) 24 families go on a picnic. 15 families bring food and 12 bring drinks. These numbers include 5 which bring both. How many families brought neither food nor drinks. U = {families} U F = {families who brought food} F D D = {families who brought drinks} 2 families brought neither food nor drinks

17 Number of Elements in a Set For each set that we are given we can count the number of elements which belong to the set. If we have set A, then n(a) is the symbol we use for the number of elements in set A. 1. If T = {2, 4, 7, 9, 11, 24} then n(t) = 6 2. If W = {odd numbers less than 25} then n(w) = If C = {1, 2, 3, 4} and D = {3, 4, 5, 6, 7} then n(c D) = 7 and n(c D) = 2

18 Number of Subsets If a set has n elements it will have 2 n subsets. 1. A = {cat, dog} A has 2 elements. A has 2 2 (=4) subsets. 2. B = {3, 5, 7} B has 3 elements. B has 2 3 (=8) subsets. 3. H = {e, h, j, n} H has 4 elements. H has 2 4 (= 16) subsets.

19 Symbols Symbol Meaning {... } the set of { } or the empty set U the universal set A c or A / n(a) union intersection is an element of is a subset of contains the complement of set A the number of elements in set A