### OPERATIONS ON SETS

(1) Union of Sets: The union of two sets A and B is the set of all the elements which are either in A or in B or in both A and B. It is denoted by \(A\cup B\). Let \(\text{A}=\left\{ 1,\,\,2,\,\,\,3 \right\}\) \(\text{B}=\left\{ 2,\,\,\,4,\,\,6 \right\}\) \(\displaystyle \therefore \text{A}\cup \text{B}=\left\{ 1,\,\,2,\,\,3,\,\,4,\,\,6 \right\}\) (2) Intersection of […]

### LAWS OF SET OPERATIONS

(1) Idempotent laws : \(\text{A}\cap \text{A}=\text{A}\) \(\text{A}\cup \text{A}=\text{A}\) (2) Identity laws: \(\displaystyle \text{A}\cap U\text{=A}\) \(\text{A}\cup \phi =\text{A}\) (3) Commutative laws: \(\text{A}\cup \text{B}\,\text{=}\,\text{B}\cup \text{A}\) \(\text{A}\cap \text{B}\,\text{=}\,\text{B}\cap \text{A}\) (4) Associative laws: \(\left( \text{A}\cup \text{B} \right)\cup \text{C}\,=\text{A}\cup \left( \text{B}\cup \text{C} \right)\) \(\text{A}\cap \left( \text{B}\cap \text{C} \right)=\left( \text{A}\cap \text{B} \right)\cap \text{C}\) (5) Distributive Laws: \(\text{A}\cup \left( \text{B}\cap \text{C} \right)=\left( […]

### CARDINAL NUMBER OF SETS

Some important results derived from Venn diagrams are as follows: A, B and C are three sets. \(\displaystyle n\left( \text{A}\cup \text{B} \right)=n\left( \text{A} \right)+n\left( \text{B} \right)-n\left( \text{A}\cap \text{B} \right)\) \(\displaystyle n\left( \text{A}\cup \text{B} \right)=n\left( \text{A} \right)+n\left( \text{B} \right)\Leftrightarrow \,\) A and B are disjoint sets. \(n\left( \text{A}-\text{B} \right)=n\left( \text{A} \right)-n\left( \text{A}\cap \text{B} \right)\) \(\displaystyle n\left( […]

### VENN DIAGRAM

The pictorial representation of sets is called Venn diagram. The universal set is represented by a rectangular region and its subsets by circles inside the rectangle. In the above given Venn diagram, \(\text{B}\subseteq \text{A}\,\)and \(\text{A}\subseteq U\) and \(\text{B}\,\subseteq \,U\). Here,\(\text{A}\subset U\)and \(\text{B}\subseteq U\). If, there is no common elements in A and B and […]

### INTERVALS

(1) Closed Interval: Let a and b are two given real numbers such that \(\text{a}\,\text{}\,\text{b}\) , then \(\left[ \text{a,}\,\text{b} \right]=\left\{ x:x\in \text{R},\,a\le x\le b \right\}\) (2) Open Interval: Let a and b are two given real numbers such that \(\text{a}\,\text{}\,\text{b,}\) then \(\left] a,b \right[\) or \(\left( a,\,\,b \right)=\left\{ x:x\in \text{R},a<x<b \right\}\) (3) Closed-open Interval: Let […]

### SUBSETS

SUBSET: A set P is said to be a subset of a set Q if each element of P is also an element of Q. If P is subset of Q, we write \(\text{P}\subseteq \text{Q}\) PROPER SUBSET: If \(\text{P}\subseteq \,\text{Q}\) and \(\text{P}\ne \text{Q}\) then P is called a proper subset of Q and we write […]

### EQUIVALENT AND EQUAL SETS

EQUIVALENT SETS Two finite sets A and B are equivalent only when number of elements of both the sets are equal. Example: A = {x : x is a natural number less than 6} B = {x : x is a vowel of English alphabet} n(A) = n(B) = 5 C = {2, 4, 6, […]

### FINITE AND INFINITE SETS

A set having definite number of elements is called finite set. In other words, we can say that the number of elements of a set is defined called finite set. A set which is not having definite number of elements is called infinite set. Example: A = {1, 2, 3, 4, 5, ……………., 100} B […]

### REPRESENTATION OF A SET

There are two types of representation: Tabular form Set – builder form Tabular or roster form: A set is described by listing elements, separated by commas, in { } brackets. Example: The set of five prime numbers can be described as {2, 3, 5, 7, 11} The set of all months having 31 days described […]

### SETS : DEFINITION

The word ‘set’ in mathematics was first used by a German mathematician George Cantor (1845 – 1918). A set is a well-defined collection of objects. Well – defined means universal truth. The elements do not varies person to person. Set is denoted by capital letters of English alphabets. E.g. A, B, C, D, F, G, […]