Representing SETS

A SET is a collection of numbers, objects or people A SET is a collection of numbers, objects or people. Each item in a set is referred to as an element or a member. Examples: - set of books for Math 436 - set of tools used by a carpenter - set of furniture for a bedroom - set of students that have completed Math 416 - set of people who were born during the month of October - set of people who work at a restaurant - set of odd numbers - set of numbers that are less than 6

There are several ways in which a SET can be expressed There are several ways in which a SET can be expressed. Three ways that will be used in this chapter are: LISTING SET-BUILDER NOTATION VENN DIAGRAM When describing a set, special brackets are used to contain the set. { } It is inappropriate to use either of the following sets of brackets to represent a set. ( ) [ ]

LISTING or Roster form - when representing a set using this form there are rules to follow. Set name must be a single uppercase letter like A, B, X or N but NOT a, b, x, Am, or Bb Each element must be separated by a comma or a semicolon. Each element must be present only once. A specific order is not required however it is often convenient to list terms in order. Suspension points (…) may be used to replace elements when there is a predictable sequence of elements.

Example of some sets by LISTING or Roster form P = {Nova Scotia, New Brunswick, Newfoundland and Labrador, Prince Edward Island} A = {Tiger Woods, Roger Federer, Thierry Henry} n(A) = 3 C = {Blue, White, Red} n(C) = 3 T = {Teddy, Mihaela, Jean, Brian, Jack, Augusta, Vicky, Mariette} n(T) = 8 M = {January, March, May, July, August, October, December} n(M) = 7 O = {1, 3, 5, 7, 9, …} n(O) = N/A E = { } n(E) = 0 S = {0, 1, 4, 9, 16, …} n(S) = N/A B = {-2, -1, 0, 1, 2, … , 6} n(B) = 9 L = {1, 4, 3, 0, 2} n(L) = 5 D = {-40, -38, -36, … , 26, 28, 30} n(D) = 36 Some of these sets are finite (countable number of elements) and some are infinite (uncountable number of elements). The number of elements in a set is referred to as the cardinal number of a set. If a set is finite, it has a cardinal number. If it is infinite, it does not. The cardinal number is denoted as shown n(set name). For the first set above it would be shown as follows: n(P) = 4

Notice that the cardinal number for set E is 0 Notice that the cardinal number for set E is 0. This set is a special set in that it has no elements. We have several names for it – Empty Set or Null Set or Void Set. Either way it has no elements and it can be represented in either of 2 ways. { } or Ø But not { Ø } For our purposes we will be working with sets involving numbers more than people or other things. There are special sets involving numbers that you need to know as they will be referred to often throughout this and other Math courses. See handout on Groups of Numbers.

SET-BUILDER NOTATION - This is basically a form where the set is described instead of listed. The description however follows a strict format. A set-builder description of the set of natural numbers less than 10 is shown below. A = { x Є N | x < 10} As with listing sets the set name is a single uppercase letter ( ‘A’ in this case). When expressing with set-builder notation there are new symbols and concepts that you must become familiar with. Membership symbol, Є - this symbol means ‘is an element of’ OR ‘belongs to’. 1 Є L 5 Є L 5 does not belong to L L = {1, 4, 3, 0, 2} 3 Є L -1 Є L -1 is not an element of L 0 Є L

A = { x Є N | x < 10} When expressing in set-builder notation, the first section inside the bracket states that ‘x Є ‘ of some universe. A universe is a set from which all possible elements are considered for the set being described. The universe is usually natural numbers (N), integers (Z) or real numbers (R). Separating the first section in the brackets from the second section, is a the following symbol ‘ | ’. This symbol can be read as ‘such that’ OR ‘whereas’. The second section in the brackets describes the elements of the set. It basically restricts the elements from the universe that satisfies the set. N = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, …} N = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, …} A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

P = {Nova Scotia, New Brunswick, Newfoundland and Labrador, Prince Edward Island} P = {x Є a province of Canada| x is an Atlantic province} C = {Blue, White, Red} C = {x Є a color| x is a color in the Montreal Canadiens jersey} T = {Teddy, Mihaela, Jean, Brian, Jack, Augusta, Vicky, Mariette} T = {x Є a PACC teacher| x is a Math teacher} M = {January, March, May, July, August, October, December} M = {x Є a month of the year| x has 31 days} O = {1, 3, 5, 7, 9, …} O = {x Є N| x is an odd number} S = {0, 1, 4, 9, 16, …} S = {x Є N| x is a perfect square} B = {-2, -1, 0, 1, 2, … , 6} B = {x Є Z| -3 < x < 7} D = {-40, -38, -36, … , 26, 28, 30} D = {x Є Z| x is an even number and -41 < x < 31} The same set can be represented either by listing or set-builder notation and if one form is presented, we should be able to convert it to the other form.

VENN DIAGRAM – A closed figure containing the elements of the set. A = {x Є Z| -4 < x < 6 and x is an even number} … Set-builder notation A = {-2, 0, 2, 4} … Roster form A 2 4 -2 Venn diagram When constructing a Venn diagram, a closed figure must be drawn with each element represented inside by a dot. The name (A in this case) must sit outside of the closed figure. U = {0, 1, 2, 3, 4, 5, 6, 7} 6 7 U A = {0, 1, 2, 3} A 2 1 3 B 4 5 3 B = {3, 4, 5}

… 8 7 N 1 9 2 A = {x Є N| 2 < x < 7} A 6 4 3 5 9 2 … A = {x Є N| 2 < x < 7} A 6 4 3 5 B = {x Є N| x = 3} B 3 U 8 7 1 2 6 4 5 3 A B C U = { } A = { } B = { } C = { }

R Q Q’ Z N

What is the difference between sets A and B? A = {x Є N| 4 < x < 7} B = {x Є N| 5 ≤ x ≤ 6} 4 5 6 7 8 4 5 6 7 8 What is the difference between sets A and B? What is the difference between sets A and C? What is the difference between sets B and D? What is the difference between sets C and D? C = {x Є R| 4 < x < 7} D = {x Є R| 5 ≤ x ≤ 6} 4 5 6 7 8 4 5 6 7 8