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