8 points you must know about Set

Theory

1. VOID, NULL OR EMPTY

SET

A set without any elements in it is called the null set or empty set and is denoted by (phi).Φ

The number of elements of a set A is denoted as n (A) and n ( ) = 0 as it contains no elements at all. Φ

Odd numbers which are divisible by 2 are well defined objects for an empty set.

2.  SINGLETON

SET

As the name signifies, the set that contains only one element is called Singleton Set.

3. FINITE AND INFINITE

SET

A set which is formed with finite number of elements, is called a finite set. Otherwise it is called an infinite set.

For example, the set of all days in a week is a finite set whereas; the set of all integers is an infinite set.

4.  UNION OF

SETS

A Union of Sets refers to the union of two or more sets that involves all elements that belong to any of these sets.

The symbol that denotes union of sets is ‘ ’ i.e. A B = ∪ ∪Union of set A and set B = {x: x  A or x  B (or both)}

Example: A = {1, 2, 3, 4} and B = {2, 4, 5, 6} and C = {1, 2, 6, 8}, then A B C = {1, 2, 3, 4, 5, 6, 8} ∪ ∪

5. INTERSECTION OF

SETS

It is the set of all the elements, which are common to all the sets.

The symbol that denotes intersection of sets is ‘∩’ i.e. A ∩ B = {x: x  A and x  B}

Example: If A = {1, 2, 3, 4} and B = {2, 4, 5, 6} and C = {1, 2, 6, 8}, then A ∩ B ∩ C = {2}

6. DIFFERENCE OF

SETS

The difference of set A to B, expressed as A – B, is the set of those elements that are in the set A but not in the set B i.e. A – B = {x: x  A and x B}∉

Example: If A = {a, b, c, d} and B = {b, c, e, f} then A–B = {a, d} and B–A = {e, f}.

For two sets A and B, symmetric difference of A and B is explained by (A – B) (B – A) and is denoted by A  B.∪

5. SUBSET OF A

SET

A set A can be called a subset of the set B if each element of the set A is also the element of the set B. The symbol used is ‘ ’ i.e. A B  (x  A => x  B).⊆ ⊆

Point to be noted is that each set is a subset of its own set.

Moreover, another interesting fact is that a void set is a subset of any set.

7. SUBSET OF A SET

(contd.)

If there is at least one element in B which is not present in the set A, then A is a proper subset of set B and is epressed as A B. e.g. If A = {a, b, c, d} and B = {b, c, d}. ⊂Then B A or equivalently A B (i.e. A is a super set of ⊂ ⊃B). Total number of subsets of a finite set containing n elements is 2n. 

Sets A and B are said to be equal if A B and B A; we ⊆ ⊆write A = B.

8. DISJOINT

SET

If two sets A and B have no common elements i.e. if no element of A belongs to B and no element of B belongs to A, then A and B are said to be Disjoint Set. Hence for Disjoint Set A and B => n (A ∩ B) = 0.  

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