SET:

A set is a collection of definite well defined objects.

A set is a collection of objects which are distinct from each other.

A set is usually denoted by capital letter, i.e, A, B, S, T, G etc.

A set elements are denoted by small letter, i.e, a, b, s, t etc.

CONSTRUCTION OF SET:

In construction of set, two methods are commonly used-

1) Roster Method (Enumeration)- In this method we prepare a list of objects forming the set, writing the elements one after another between a pair of curly brackets.

For example:

A = {a, b, c, d}.

2) Description Method- In this method we describe the set in symbolic language.

For example:

A set of integer numbers which is divisible by 3 is written as,

A = {x : x is an integer divisible by 3}

TYPES OF SET:

1) Singleton set- If a set consisting only 1 element is known as singleton set.

For example:

A = {a}.

2) Finite set- If a set consisting finite number of elements is known as finite set.

For example:

A = {2, 4, 6, 8}.

3) Infinite set- If a set consisting infinite number of elements is known as infinite set.

For example-

The set of all natural numbers.

A = {1, 2, 3,......}

4) Equal sets- Two sets A and B consisting of the same elements is known as equal set.

For example:

A = {a, b, c, d} and

B = {a, b, c, d}

5) Empty set: If a set consisting no elements is known as empty set or null set or void set.

For example:

A = { ∅ }

6) Subset- Suppose A is a given set, and any set B exist exist whose elements are also an element of A,than B is called subset of A.

For example:

A = {1, 2, 3, 4, 5, 6, 7, 8} and B = {2, 4, 6, 8}

Than, B ⊆ A. (read as B is the subset of A)

Now take another example;

A = {1, 2, 3, 4, 5, 6, 7, 8} and B = {1, 2, 3, 4, 5, 6, 7, 8}

Than, B ⊆ A. (read as B is the subset of A)

7) Proper Subset- If B is the subset of A, and B≠A, then B is proper subset of A.

For example:

A = {1, 2, 3, 4, 5, 6, 7, 8} and B = {2, 4, 6, 8}

Than, B ⊂ A. (read as B is the proper subset of A)

8) Power set- The set of all subset of a set A, is known as power set of A.

For example:

A = {a, b, c}

Than

Power set,P(A) = {{∅ }, {a}, {b}, {c}, {d}, {ab}, {ac}, {ad}, {bc}, {bd}, {cd}, {abc}}

Problems based on above study:

Prob. If A, B and C are three sets, then prove that-

Prob. If A and B are two sets, then prove that-

(A-B) ∪ (B-A) = (A ∪ B) - (A ∩ B)

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