Definition and Representation A set is a well-defined collection of objects; The objects are called elements or members of the set; A set can be represented by; (a) A description in words: A = (the first 5 natural numbers) (b)listing all its elements or members: A = {1, 2, 3, 4, 5 } (c) Set –builder notation: A = { x: 1 ≤ x ≤ 5, x  R }

A set can also be represented diagrammatically by means of a Venn diagram:  E RRRR

An element of a set is denoted by the symbol  4  {2,4, 6, 8, 10 } A non–element of a set is denoted by the symbol 7  { 1, 2, 3,4, 5 } The number of elements in a set A is written as n (A) A = {1, 2, 3, 4, 5} Hence n( A ) = 5 Elements of a set

Two sets are said to be equal or identical if both sets have exactly the same elements which may not be shown in the same order. e.g. { a, b, c, d } = { c, d, b, a } Equal Sets

Empty Set An empty or null set has no element. It is denoted by the symbol Ø or { } e.g. (a) { dogs with 3 pairs of legs } = Ø e.g. (b) { months with 29 days in 2003 } = Ø

A universal set contains all the elements It is denoted by the letter ‘  ‘ e.g. (a) The universal set of integers  = {…,-2, -1, 0, 1, 2,…} (b) The universal set of animals  = { hens, dogs, lions, cats,…} Universal Set

Finite and Infinite sets A finite set has a limited number of elements: e.g. { even numbers between 11 and 20 } is equal to { 12, 14, 16, 18 } An infinite set has an unlimited numbers of elements: e.g. { odd numbers } = { 1, 3, 5, 7, 9,…}

The complement of a set A is the set of those members in the universal set that are not members of A. The complement of the set is denoted by A’. e.g. Given that  = {1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {1, 5, 8, 9 } list the numbers of the set A’. Illustrate the set A’ by means of a Venn diagram. Complement Set

Solution:  = {1, 2, 3, 4, 5, 6, 7, 8, 9 } A = { 1, 5, 8, 9, } Hence A’ = { 2, 3, 4, 6, 7 }  1, 5, 8, 9 A 2, 3, 4, 6, 7

If two sets A and B have no common element, they are said to be disjoint. e.g. If A = { even numbers } andB = { odd numbers } then A and B are disjoint Disjoint Set

Subset The set A is a subset of the set B if every element of A is an element of B. Subset is denoted by the symbol  : A = { 3, 5 } B = { 1, 3, 5, 7, 9, 11} A  B

A subset can be illustrated by means of a Venn diagram: Every set is a subset of itself: {a, b, c, }  { a, b, c} EEEE 3333 EEEE e

Intersection

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n(A  B) AB

Union

n(A  B) = n(A) + n(B) – n(A  B) AB

History George Cantor, ( ), German mathematician – founder of “The Set Theory” Set the foundations of many advanced mathematical works. Concepts of sets can help in classifying & counting things. Make learning of Mathematics more meaningful

Real Life Examples  A set is a collection of things. Some examples: A coin collection The English alphabet Even numbers Odd numbers

Hands-on Activities  Get students to sort items by colour, size, and shape. Some examples Things to sort by colour: crayons, markers, backpacks, clothes Things to sort by shape: tables, block Things to sort by size: books, shoes, students

Buttons Divide the class into groups of three or four students each. Give each group a small container of assorted buttons. Ask each student to sort the buttons according to colours, shape & size.