# Discrete Mathematics - Sets and Types Of Sets

## Sets and Types Of Sets

**Question: What is a Set?**

Answer: A collection of well defined objects are called set. Upper case letters are used to denote a set.

Some examples of set are

(i) A = {I, 3, 5, 7, ... ,99},

(ii) B = {I, 4, 9,16,25, ... },

(iii) B = {x : x is a real number satisfying x(x²-4) = O}

**Question: Identify which of the following are examples of set?**

⒜ Collection of all chairs and tables of a class room

⒝ All tall men of a class

⒞ All students of BCA class

⒟ All good students of a college.

⒠ Collection of Indian Prime Ministers of age more than 150 years.

Answer: ⒜, ⒞, ⒠ are examples of set. ⒠ is a null set.

**Question: What is a Null set or empty set?**

Answer: A set having no element is called null set or empty set.

It is generally denoted by the symbol ∅.

For example:

∅ = {x : x is a real number satisfying x²+ 4 = 0}

**Question: What is a Singleton set or unit set?**

Answer: A set having only one element is called a unit set or a singleton.

e.g. S = {9}

**Question: What is a pair set?**

Answer: A set which contains only two elements is known as a pair set. Consider the examples

S = {e, f}

S = {{a}, {1, 3, 5}}

**Question: Define Finite set and infinite set:**

Answer: A set is called finite if it be empty or contains a finite number of elements, otherwise a set is called infinite.

Example of Finite set: A set containing even integers between 2 and 20

Example of Infinite set: A set containing even integers.

**Question: What is a sub-set? What is a super-set?**

Answer: If every element of a set A be also an element of another set B, then A is called a sub-set of Band B is called super set of A.

Symbolically, A ⊆ B or B ⊇ A.

If B contains some elements which are not elements of a A, then A is a proper sub-set of B. Or B is proper super-set of A.

i.e. A ⊂ B or B ⊃ A

**Question: What are Disjoint sets?**

Answer: Two sets A and B are called disjoint if they have no element in common.

For example, A = {1, 2, 3} and B = {10, 11, 12} are disjoint.

**Question: When two sets are said to be identical or equal?**

Answer: Two sets A and B are said to be equal or identical, i.e., A = B, if they have the same elements. In this case, all elements of A are also the elements of B.

For example, A = {I, 2, 3} and B == {3, 1, 2} are equal.

**Question: What is a Universal set?**

Answer: A universal set is the set of all elements from which elements may be chosen to form sets for a particular problem. It is generally denoted by S or U.

e.g.

U = {a, b, c, d, e, ..., z}

**Question: What do you mean by Cardinality of a set?**

Answer: If S be a set, then the number of elements present in the set S is known as cardinality of S and is denoted by |S|. Mathematically if S = {s1, s2, s3, ..., sk}, then |S| = k; k ∈ N.

For example, A = {2, 4, 8, 16, 32, 64, 128, 256}

∴ |A| = 8

**Question: When the two sets (say A and B) are said to be comparable?**

Answer: Two sets A and B are said to be comparable if any one of the following relation holds.

i.e.

(i) A ⊂ B or

(ii) B ⊂ A or

(iii) A = B.

**Question: What is power set?**

Answer: If A be a set, then the set of all subsets of A is known as power set of A. Which is denoted

by P(A).

Mathematically, P(A) = {X : X ⊆ A}

e.g.

A = {1, 2}

P(A) = {{1}, {2}, {1, 2}, ∅}

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