INTRODUCTION OF SETS AND PROPERTIES OF SETSSets.pptx

Introduction of sets and properties of sets.

1. Sets M.MONICA SREE PE D AGOGY OF MATHEMATICS

2. INTRO D UCTION A set is a collection of well-defined objects . The set containing no elements is called an empty set or a void set. It is usually denoted by ∅ or { }. By A ⊆ B, we mean every element of the set A is an element of the set B. In this case, we say A is a subset of B and B is a super set of A. For any two sets A and B , if A ⊆ B and B ⊆ A, then the two sets are equal.

3. UNION AN D INTERSECTION Union of two sets A and B is denoted by A ∪ B and is defined as A ∪ B = {x : x ∈ A or x ∈ B} and the Intersection of two sets A and B is denoted by A ∩ B and is defined as as A ∩ B = {x : x ∈ A and x ∈ B}.

4. Power set If A is a set, then the set of all subsets of A is called the power set of A and is usually denoted as P(A). That is, P(A) = {B : B ⊆ A }. The number of elements in P(A) is 2n, where n is the number of elements in A.

5. COMPLEMENT AN D SET D IFFERENCE The complement of A with respect to U is denoted as A ' and defined as A ' = {x : x ∈ U and x /∈ A}. The set difference of the set A to the set B is denoted by either A − B or A /B and is defined as A − B = {a : a ∈ A and a /∈ B}.

6. PROPERTIES OF SET OPERATIONS Commutative ( i ) A ∪ B = B ∪ A ( ii) A ∩ B = B ∩ A. Associative ( i ) (A ∪ B) ∪ C = A ∪ (B ∪ C) ( ii) (A ∩ B) ∩ C = A ∩ (B ∩ C). Distributive ( i ) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) ( ii) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C ).

7. Identity A ∪ ∅ = A (ii) A ∩ U = A. Idempotent A ∪ A = A (ii) A ∩ A = A. Absorption A ∪ (A ∩ B) = A (ii) A ∩ (A ∪ B) = A.

8. De Morgan Laws ( i ) (A ∪ B) ' = A ' ∩ B ' (ii) (A ∩ B) ' = A ' ∪ B ' (iii) A − (B ∪ C) = (A − B) ∩ (A − C) (iv) A − (B ∩ C) = (A − B) ∪ (A − C). On Symmetric Difference A∆B = B∆A (ii) (A∆B)∆C = A∆(B∆C) (iii) A ∩ (B∆C) = (A ∩ B)∆(A ∩ C).

9. CARTESIAN PRODUCT The set of all ordered pairs of elements. The Cartesian product of two sets is a set of ordered pairs , Then the Cartesian product of A an d B is denoted by A × B is defined by A × B = {(a, b) : a ∈ A, b ∈ B}. The Cartesian product of three sets is a set of ordered triplets , let A, B and C be three sets. Similarly , the Cartesian product A × B × C is defined by A × B × C = {(a, b, c) : a ∈ A, b ∈ B, c ∈ C }.

10. NON EMPTY SET For non-empty sets A × B = B × A if and only if A = B. We know that R denotes the set of real numbers and R × R = {(x, y) : x, y ∈ R }. R × R can be represented as R2. Note that R × R is a set of ordered pairs Then R × R × R = {(x, y, z) : x, y, z ∈ R}. and R × R × R as R3. It is known as R × R × R is a set of ordered triplets .