Sets, Relations And Mappings

About Myself: My Name -

Rajni Handa

Working as - Maths Mistress At - Govt. Girls Sen. Sec. School, Rajpura Town

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To give the knowledge of sets, the need and value of numbers as applied to everyday life.

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To build confidence and mathematical procedures.

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To impart an understanding of mathematical concepts and their applications, which are basic to further studies in mathematics.

skill

involving

The Set theory was developed by a German mathematician George Cantor. Sets provide a useful way of representing groups of things and correlating them.

George Cantor (1845-1918)

- SET- A collection or aggregate of welldefined objects is called a set. - The objects of a set are called its elements or members. -

If x is a member of the set A, then we write x є A (x belongs to A).

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e.g. (a) The set of natural numbers less than hundred. (b) The set of pages in a book.

(1) Tabular Form [Roster Notation] In this form, we enumerate or list all the elements. e.g. (i) W (the set of whole numbers) = {0,1,2,3…} (ii) Z (the set of integers) = {……, -2,-1,0,1,2,3….} (iii) N (the set of natural numbers)

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The Rule Method [Set Builder Notation] In this form, we specify the 'defining property‘, i.e. A = {x: x has property p} e.g. (i) W = {x: x is a whole number} (ii) Z = {x: x is an integer} (iii) N = {x: x is a natural number}

The Empty Set (ø) or { } – A set which contains no element is called the empty set. - e.g. A = {x: x is a married bachelor} = ø - B = {x: x+1 = 0, x є N} = ø

Finite Set – A set with finite elements is finite set

called a

- e.g. A = {1,2,3,4,5…….50} - B = {x: x is a school in India} Infinite Set – A set which is neither the null set nor a finite set is called an infinite set. - e.g. A = {1,2,3,4,5…….} - B = {x: x є W}, where W denotes a set of all whole numbers

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Disjoint Sets Two sets are disjoint, if they have no element in common. e.g. if A = {1,2,3}, B = {4,5,6}, then A and B are disjoint, since there is no element common in A and B.

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Equal Sets

Two sets are said to be equal, if they contain the same elements. - e.g. if A = {1,2,3}, B = {1,2,3}, then A=B (or A ↔ B) - If A = {x: x is a letter of the word TALENT} B = {x: x is a letter of the word LATENT}, then A = B.

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Equivalent Sets

Two sets are equivalent, if and only if, a one-to-one correspondence exists between them. e.g. (a) If A = {1,2,3}, B = { a,b,c}, then A and B are equivalent, since the correspondence between the elements of A and B is one-toone. (b) If A = {x: x є N, x < 5}, B = {x: x is a letter of the word DEAR}; then A and B are equivalent.

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Universal Set (ξ) A set which contains all sets under consideration as sub-sets. It is denoted by ξ or X or U or [ ].

e.g. Let A = {1,2,3}, B = {2,3,4}, then the universal set ξ might be {1,2,3,4,5,6} or {x: x є N}, or {x: x є W}

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Complement Set The complement of a set A is the set of elements which do not belong to A. It is denoted by A'

e.g. (a) If A = {1,2,3,4}, ξ = {1,2,3,4,5,6,7,8}, then A' = {5,6,7,8} (b) If B = {x: x is a book on algebra in your library}, then B' = {x: x is a book in your library and x Є B}

The complement of a set may be represented by a Venn diagram as:

SUB SET - Set A is a subset of B, if every element of set A is also an element of set B. It is denoted by A B. - e.g. A = {1,2}, B = {1,2,3}, then A B.

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Power Set It is the set of all subsets of a set. If A = {1, 2}, then the power set of A =

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{ {1}, {2}, {1, 2}, { } }

Cardinal Number of a Set The cardinal number of a finite set 'A' is the number of elements of the set A. It is denoted by n(A) If A = {1, 2, 3}, then n(A) = 3

(1) Intersection of two Sets: A ∩ B = {x: x є A and x є B}. This can be represented as:

(2) Union of two sets: A U B = {x: x є A alone or x є B alone or x є A ∩ B} More often, we write A U B = {x: x є A or x є B}. This is represented as:

(3) Difference of two sets: The difference of two sets A and B is the set A – B, in which elements do not belong to B, but belong to A. It may be represented by a Venn diagram as:

(1) Write the following in tabular form: (i) {x: x is a vowel in the word MATHEMATICS} (ii) {x: x is a consonant in the word NOTATION} (iii) {x: x = 2n; n є N} (iv) {x: x = 2n + 1; n є N} (v) {x: x -1 = 0} (vi) {x: 2x – 1 =0} (vii) {x: x = prime number and x ≤ 7, x є N}

(2) Write the following in set builder notation: (i) A = {1, 4, 9, 16, 25, 36, 49} (ii) B = {1, ½, 1/3, ¼, 1/5, 1/6, 1/7} (iii) C = {2, 3, 5, 7, 11, 13, 17, 19} (iv) D = {7, 14, 21, 28, 35, 42, 49} (v) E = {1, 8, 27, 64, 125} (vi) F = {2, 6, 10, 14, 18, 22, 26} (vii) G is the set, whose elements are obtained by adding 1 to each of the even numbers.

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What are infinite things?

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What are finite things?

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What are whole numbers?

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What is a set?

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What is intersection of a set?

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What is union of sets?

• I would like to thank ……………… for his constant encouragement throughout this training period. • I would also like to thank Microsoft Corp. for their excellent training material. • Books consulted: Mathematics Book for Class IX-X by ICSE • Website: www.wikipedia.org