Sets

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Mashallah Tuition Centre & Pre-Entry Test Centre. MATHEMATICS -1

Chapter - 1

SETS Definition of set A collection well defined and distinct objects such as number, points, shapes ideas etc is called a set which is denoted by Capital letters of Alphabets i.e. A, B, C… X, Y, Z. Its symbol is { }.

Notations of Sets 1. Tabular Method In this method, we define a set merely by listing its elements in closed within braces { }. Example: N = {1, 2, 3…} 2. The Descriptive Method In this method the element of the set are described by stating their common characteristic which an object must posses in order to be an element of the set. Example: Set of Odd Number (O) A = {x | x is an odd integer} 3. Set builder Notation In this method the elements of the set have to satisfy some condition i.e. {x | x satisfy some condition} Example: N = {x | x ∈ N} If A be the set of rational numbers, then in Set Builder Notation we will write it as A = {x | x is a rational number} 4. Venn Diagram In this method the set is written in figure such as rectangle, square, circle or any other. Example: Draw a Venn Diagram to represent. U = {1, 2, 3, … ,10} and A = {1, 3, 5, 7, 9}

Figure U A 3

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Kinds of Sets Universal sets (U) Universal set is the set, which contains all the available elements.

Equal Sets Equal set have exactly the same number of same elements. For example set A = {a,b,c,d} and B = {b,c,a,d} are equal since they have the same elements. We write A = B, if A and B are equal.

Equivalent Sets Two sets A and B are said to be equivalent, if they have same number of elements, it is denoted by A ∼ B. Example: A = {a,b,c} , B = {1,2,3} Then A ∼ B, because order of A is equal to order of B

Disjoint Sets If two sets do not have an element in common, they are said to be disjoint as A∩B =φ

Finite Sets

A set is finite if it contains a limited number of different elements. Example: A = {1, 2, 3, 4}, B = {a, b, c, d, e, f} etc.

Infinite Sets

A set which is not finite is called infinite set. Example: A = {1, 3, 5 …}, B = {1, 2, 3 …}, C = {… -3,-2,-1,0,1,2,3 …}

Null Or Empty Set

A Null set { } or empty set is the set which contains no element. Example: A = {x | x > 5 and x < 2} = ∅

Power sets The all of possible subset of a set A is called the power set of A and it is denoted by the symbol P(A). Example: If A = {1,2,3} then, P(A) = {∅ {1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}} Note: The number of subsets of power set can be obtained by using 2m where m is no. of elements in set A.

Subset

If every element of a set A is also an element of a set B, then A is a subset of B and we write A ⊆ B. Example: If A = {1,2,3} and B = {1,2,3,4,5} then A ⊂ B

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Superset

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If B is a subset of a set A, then A is called a superset of B, denoted by A ⊂ B or B ⊃ A Example: If A = {1,2,3} and B = {1,2,3,4,5} then B ⊃ A

Proper Subset If A is subset of B and A ≠ B then we write A ⊂ B and say that A is Proper subset of B. Example: If A = {1,2,3} and B = {1,2,3,4,5} then A ⊂ B

Improper Subset If A ⊆ B and B ⊆ A, then sets A and B are said to be improper subsets of one another if A=B Example: If A = {1,2,3} and B = {1,2,3} then A ⊆ B or B ⊆ A

Set Operation Complement The complement of a set A relative to a universal set U is the set if all elements in U except those in A denoted by A’ and A’ = U – A. A’ = {x | x ∈ U and x ∉ A}

Exhaustive Sets

If A and B be subsets of a set U such that, A U B = U

Symmetric Difference

The symmetric difference of sets A and B, denoted by A ∆ B, is the set containing those elements which are either in A or in B but not in both A and B. Example: If A = {1,2,3,4,6} and B = {1,3,5,7} then A ∆ B = {2,4,5,6,7} Note: A ∆ B = A U B – A ∩ B

Difference of Two Sets The difference of sets A and B, denoted by A-B or A/B, is the set containing those elements that are in A but not in B. A - B = {x | x ∈ A and x ∉ B}

Intersection of Sets

The intersection of two sets A and B Is the set of element which are common to both A and B it is denoted by A ∩ B. A ∩ B = {x | x ∈ A and x ∈ B}

Union of Sets

The union of two sets A and B is set of elements which are in A or B or both it is denoted by A ∪ B. A U B = {x | x ∈ A or x ∈ B}

Number of Elements n (A) n (Ф) = 0 n (A U B) = n (A) + n(B) – n(A ∩ B) n (A ∩ B) = n (A) + n(B) – n(A U B) n (A’) = n(U) – n(A)

The Cartesian product of two Sets The Cartesian production of any set A with other set B is the set all ordered pairs (a, b) where a ∈ A and b ∈ B it is denoted by A x B = {(a, b) | a ∈ A, b ∈ B}.

Some Important Conditions 1. 2. 3. 4.

A ⊆ B and B ⊆ A ⇒ A = B if A ⊂ B then B ⊃ A A = B and B = C ⇒ A = C A ∼ B and B ∼ C ⇒ A ∼ C

5. if A = { } = φ then P(A) = {A} 6. the number of subset of A i.e. n{P(A)} is 2m where m is number of elements in A i.e. n(A) = m.

Some Important Laws 1. Idempotent Law AUA=A A∩A=A 2. Associative Laws If A, B and C are any three sets then (A U B) U C = A U (B U C), (A ∩ B) ∩ C = A ∩ (B ∩ C) 3. Distributive Laws If A, B and C are any three sets then A ∩ (B U C) = (A ∩ B) U (A ∩ C) (B U C) ∩ A = (B ∩ A) U (C ∩ A) A U (B ∩ C) = (A U B) ∩ (A U C) 4. Identity Law AUφ=A A∩U=A AUU=U A∩φ =φ 5. Complement Law A U A’ = U A ∩ A’ = φ (A’)’ = A U’ = φ , φ’ = U 6. Commutative Law A U B = B U A, A∩B=B∩A 7. De-Morgan’s Law If A, B and C are any three sets then (A U B)’ = A’ ∩ B’ (A ∩ B)’ = A’ U B’

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Notation for Sets of Numbers N = {1, 2, 3, …} i.e. the set of all natural numbers. W = {0, 1, 2, 3…} i.e., the set of all non-negative integers. Or set of whole numbers. Z = {…., -3 , -2, -1, 0, 1, 2, 3,…} i.e., the set of all integers. P = {2, 3, 5, 7, 11 …} i.e. the set of all Positive Prime numbers. O = {±1, ±3, ±5 …} i.e. the set of all Odd numbers. E = {0, ±2, ±4, ±6 …} i.e. the set of all Even numbers. Q = {x|x = p/q, p and q ∈ Z, q ≠ 0} i.e., the set of all rational numbers. Q’ = I = { x|x ≠ p/q, p and q ∈ Z, q ≠ 0} i.e., the set of all irrational numbers. R = { x|x = p/q, x ≠ p/q, p and q ∈ Z, q ≠ 0} i.e., the set of all real numbers. Or R = Q U Q’

Rational Number

Rational Number is a number which can be expressed as a terminating decimal fraction or a recurring decimal fraction.

Irrational Number

Irrational Number is a number which can be expressed only as a non-recurring, nonterminating decimal fraction.

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Set of Real Numbers Union of the sets of rational numbers (Q) and irrational numbers (Q’) is called the set of real numbers, denoted by R. i.e. R = Q U Q’ or R = {x | x ∈ Q or x ∈ Q’} The sets Q and Q’ are disjoint sets.

Properties of Real Numbers Properties w.r.t. to Addition

1. Closer property Sum of any two real numbers is also a real number. i.e. x, y ∈ R ⇒ x+y∈R 2. Commutative Property x + y = y + x , ∀ x, y ∈ R (∀ is read as “for all”) 3. Associative property x + (y + z) = (x + y) + z ,

∀ x, y, z ∈ R

4. Additive Identity There exists a number 0 ∈ R such that x + 0 = 0 + x = x, ∀x∈R The element 0 is called the additive identity. 5. Additive Inverse For every x ∈ R, there exists an element x’ ∈ R such that x + x’ = 0 = x’ + x x’ is called Additive Inverse of x and is denoted by “-x”

Property w.r.t. Multiplication 1. Closer property Products of any two real numbers is also a real number. i.e. x, y ∈ R ⇒ x.y ∈ R 2. Commutative Property x.y = y.x , ∀ x, y ∈ R 3. Associative property x (y.z) = (x.y) z , ∀ x y ∈ R 4. Multiplicative Identity There exists a number 1 ∈ R such that

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x . 1 = 1 . x = x, The number 1 is called the Multiplicative identity. 5. Multiplicative Inverse For each x ∈ R, x ≠ 0, there exists an element x* ∈ R such that x . x* = x* . x = 1 x* is called Multiplicative Inverse of x. It is also written as 1/x or x-1 Thus, x. 1/x = x/x = 1/x .x = 1 or x . x-1 = x-1.x = 1 Note: The Multiplicative Inverse of x-1 is x,

i.e. (x-1)-1 = x,

∀ x ∈ R, x ≠ 0

Distributive property of Multiplication w.r.t. Addition x (y + z) = xy + xz and (y + z) x = yx +zx ∀ x, y, z ∈ R Trichotomy Property For any two real numbers x and y, either x < y or x = y or x > y.

Property of Equality of real Numbers In the set R of real numbers, a relation of equality to be denoted by “=” is defined. This relation satisfies the following properties: 1. Reflexive Property x = x, for all x ∈ R 2. Symmetric property x = y ⇒ y = x, for all x, y ∈ R 3. Transitive property x = y and y = z ⇒ x = z for all x, y, z ∈ R 4. Additive property x = y ⇒ x + z = y + z and z + x = z + y, for all x, y, z ∈ R i.e. addition of same no. to each side of an equality does not change the relation. 5. Multiplicative Property ∀ x, y, z ∈ R x = y ⇒ xz = yz (Right Multiplication) and zx = zy (Left Multiplication) i.e. Multiplication by the same no. to each side of an equality does not change the relation. 6. Cancellation Property w.r.t. Addition ∀ x, y, z ∈ R (a) x + z = y +z ⇒ x = y (Right Cancellation ) (b) z + x = z + y ⇒x=y (Left Cancellation) 7. Cancellation Property w.r.t. Multiplication ∀ x, y, z ∈ R, z ≠ 0 (a) xz = yz ⇒ x = y (Right Cancellation ) (b) zx = zy ⇒ x = y (Left Cancellation)

Note: Additive property and Cancellation property w.r.t. Addition are converse of each other. Similarly, Multiplicative property and Cancellation property w.r.t. Multiplication are converse of each other.

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