# Introduction to Set

A well-defined collection of objects is called at set.

**The types of Representation are:**

(i) Roaster form

(ii) Set builder form

# Types of Set

**1. ****Subset – **Let A and B be two sets such that every element of A is in B, then we say that A is a subset of B and we write, A Í B.

**2.** **Equal sets –** Two sets A and B are said to be equal, if every element of A is in B and every element of B is in A, and we write, A = B.

**Note 1** – Repetition of elements, in a set does not affect the equality of sets e.g. {1, 2, 3, 1, 2, 3,…) = {1, 2, 3}

* Note 2 – *The order in which the elements of a set are listed is immaterial e.g.

**3.** **Proper subset – **If A Í B and A ¹ B, then A is called a proper subset of B and we write, A Ì B.

**4. ****Power set – **The set of all subsets of a given set A is called the power set of A, to be denoted by P (A).

**5.** **Singleton Set– **A set containing one element is called Singleton Set.

**6.** **Universal Set –** The set which contains all the elements under consideration in a particular problem is called the universal set denoted by S. Suppose that P is a subset of S. Then the complement of P, written as P^{c} (or P’) contains all the elements in S but not in P. This can also be written as S – P or S ~ P.

**7.** **Equivalent Set –** Two finite sets A & B are said to be equivalent if n (A) = n (B).

**Note: **Equal sets are equivalent but equivalent sets need not be equal.

**8.** If A contains n elements, then the number of proper subsets of A is (2^{n} –1).

**9. **If A contains n elements, then the number of subset of A = 2 ^{n}.

**10.** **Cardinal number – **The number of distinct elements contained in a finite set A is called its cardinal number. It is denoted by n (A).

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# Operations on sets

[Symmetric difference]

# Laws of operations

# De-Morgan’s Laws

**Important ****results:**

# Cartesian product of sets

**Results on Cartesian product of sets:**

(iv) If A and B have *n *elements in common, then A ´ B and B ´ A have n^{2} elements in common.

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