Important Terms, Definitions & Formulae 01. TYPES OF INTERVALS a) Open interval : If a and b be two real numbers such that a b then,...

**Important Terms, Definitions & Formulae**

**01. TYPES OF INTERVALS**

**a) Open interval : If a and b be two real numbers such that a b then, the set of all the real numbers**

**lying strictly between a and b is called an open interval. It is denoted by ] a , b[ or a , b i.e.,**

**x R : a x b .**

**b) Closed interval : If a and b be two real numbers such that a b then, the set of all the real numbers lying between a and b such that it includes both a and b as well is known as a closed interval. It is denoted by a , b i.e., x R : a x b .**

**c) Open Closed interval : If a and b be two real numbers such that a b then, the set of all the real numbers lying between a and b such that it excludes a and includes only b is known as an open closed interval. It is denoted by a , b or a , b i.e., x R : a x b .**

**d) Closed Open interval : If a and b be two real numbers such that a b then, the set of all the real numbers lying between a and b such that it includes only a and excludes b is known as a closed open**

**interval. It is denoted by a , b or a , b i.e., x R : a x b .**

**RELATIONS**

**Defining the Relation : A relation R, from a non-empty set A to another non-empty set B is mathematically defined as an arbitrary subset of A B . Equivalently, any subset of A B is a relation from A to B.**

**Thus, R is a relation from A to BR A × B**

**Ra , b : a A, b B .**

**Illustrations:**

**a) Let A 1, 2, 4 , B 4, 6 . Let R (1, 4), (1, 6), (2, 4), (2, 6), (4, 6) . Here R A × B and therefore**

**R is a relation from A to B.**

**b) Let**

**A 1, 2, 3 , B**

**2, 3 , 5, 7 . Let**

**R (2, 3), (3, 5), (5, 7) .**

**Here**

**R A B**

**and therefore R is**

**not a relation from A to B. Since**

**(5, 7)**

**R**

**but**

**(5, 7)**

**A B .**

**c) Let A1,1, 2 , B 1, 4, 9,10 . Let a R b means a 2 b then, R ( 1,1), (1,1), (2, 4) .**

**Note the followings:**

**A relation from A to B is also called a relation from A into B. ( a , b) R is also written as aRb (read as a is R related to b).**

**Let A and B be two non-empty finite sets having p and q elements respectively.**

**Then**

**n A B n A .n B pq . Then total number of subsets of A B**

*2*^{pq}**. Since each subset of A B is a relation from A to B, therefore total number of relations from A to B is given as**

*2*^{pq}**.**

**03. DOMAIN & RANGE OF A RELATION**

**Domain of a relation : Let R be a relation from A to B. The domain of relation R is the set of all**

**those elements a A such that ( a , b) R for some b B . Domain of R is precisely written as Dom.( R) symbolically.**

**Thus, Dom.(R) a A : a , b R for some b B .**

**That is, the domain of R is the set of first component of all the ordered pairs which belong to R.**

**Range of a relation: Let R be a relation from A to B. The range of relation R is the set of all those elements b B such that ( a , b) R for some a A .**

**Thus, Range of R b B : a , b R for some a A .**

**That is, the range of R is the set of second components of all the ordered pairs which belong to R.**

**Codomain of a relation : Let R be a relation from A to B. Then B is called the codomain of the relation R. So we can observe that codomain of a relation R from A into B is the set B as a whole.**

**TYPES OF RELATIONS FROM ONE SET TO ANOTHER SET**

**Empty relation : A relation R from A to B is called an empty relation or a void relation from A to B if R Ï†**

**Universal relation : A relation R from A to B is said to be the universal relation if R A B .**

**RELATION ON A SET & ITS VARIOUS TYPES**

**A relation R from a non-empty set A into itself is called a relation on A. In other words if A is a non-empty set, then a subset of A A A2 is called a relation on A.**

**Illustrations :**

**Let A 1, 2, 3 and R (3,1), (3, 2), (2,1) . Here R is relation on set A.**

**Identity relation : A relation R on a set A is said to be the identity relation on A if**

**R ( a , b ) : a A, b A and a b .**

**Thus identity relation R ( a , a ) : a A .**

**The identity relation on set A is also denoted by I**

_{A}

**Reflexive relation : A relation R on a set A is said to be reflexive if a R a a A i.e.,**

**(a , a ) R a A .**

**NOTE The identity relation is always a reflexive relation but the opposite may or may not be true. As shown in the example above, R1 is both identity as well as reflexive relation on A but R2 is only reflexive relation on A.**

**Symmetric relation : A relation R on a set A is symmetric**

**a , b Rb , a R a , b A i.e., a R b b R a (i.e., whenever a Rb then, b Ra ).**

**Transitive relation : A relation R on a set A is**

**b , c Ra , c R i.e., a R b and b R c a R c .**

**Equivalence relation : Let A be a non-empty set, then a relation R on A is said to be an equivalence relation if**

**(i) R is reflexive i.e. ( a , a) R a A i.e., a Ra .**

**(ii) R is symmetric i.e. a , b Rb , a R a , b A i.e., a Rb b Ra .**

**(iii) R is transitive i.e. a , b R and b , c Ra , c R a ,b, c A i.e., a Rb and**

**b Rc a Rc .**

**For example, let A 1, 2, 3 , R (1, 2), (1,1), (2,1), (2, 2), (3, 3) . Here R is reflexive,**

**symmetric and transitive. So R is an equivalence relation on A.**

**Equivalence classes : Let A be an equivalence relation in a set A and let a A . Then, the set of all those elements of A which are related to a , is called equivalence class determined by a and it is denoted by a . Thus, a b A : a, b A .**

**INVERSE RELATION**

**Let R A B be a relation from A to B. Then, the inverse relation of R,**

**to be denoted by R 1 , is a relation from B to A defined by R 1 ( b , a ) : ( a , b) R .**

**Thus (a , b) R (b, a ) R 1 a A, b B .**

**Clearly, Dom. R 1 Range of R, Range of R 1 Dom. R .**

**1**

**Also, R 1 R .**

**For example, let A 1, 2, 4 , B 3, 0 and let R (1, 3), (4, 0), (2, 3) be a relation from A to B then,**

**R 1 (3,1), (0, 4), (3, 2) .**