Relation-Function Grade 12 concepts and examples
RELATION IN A SET
A relation R in a set A is a subset of A ́ A.
Thus, R is a relation in a set A Û R Í A ́ A.
If(a, b) ÎR then we say that a is related to b and write, a R b. If(a, b) ÏR then we say that a is not related to b and write, a R b.
Example Let A = {1, 2, 3, 4, 5, 6} and let R be a relation in A, given by R = {(a, b) : a - b = 2}.
Then, R = {( 3, 1), (4, 2), (5, 3), (6, 4)}. Clearly, 3 R1, 4R 2, 5 R 3and6R 4. But, 1 R 3, 2 R 4, 5 R 6, etc.
DOMAIN AND RANGE OF A RELATION
Let R be a relation in a set A. Then, the set of all rst coordinates of elements of R is called the domain of R, written as dom
(R) and the set of all second coordinates of R is called the range of R, written as range (R).
\ dom ( R) = {a : ( a , b) Î R} and range ( R) = {b : ( a , b) Î R}.
Example Let A ={1, 2, 3, 4, ..., 15, 16} and let R be a relation in A, given by
R = {(a, b) : b = a2}.
Then, R = {(1, 1), (2, 4), (3, 9), (4, 16)}.
\ dom (R) = {1, 2, 3, 4} and range (R) = {1, 4, 9, 16}. Some Particular Types of Relations
EMPTY RELATION (Or VOID RELATION) A relation R in a set A is called an empty relation, if no element of A is related to any element of A
and we denote such a relation by f.
Thus, R = f Í A ́ A.
Example Let A= {1, 2, 3, 4, 5} and let R be a relation in A, given by R = {(a, b) : a - b = 6}.
Clearly, no element (a, b) ÎA ́ A satis es the property a - b = 6. \ R is an empty relation in A.
UNIVERSAL RELATION A relation R in a set A is called a universal relation, if each element of A is related to every element of A.
Thus, R =(A ́A) Í (A ́A) is the universal relation on A. Example Let A = {1, 2, 3}. Then,
R =(A ́A) ={(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)} is the universal relation in A.
IDENTITY RELATION The relation IA = {(a, a) : a ÎA} is called the identity relation on A.
Example Let A = {1, 2, 3}. Then,
IA = {(1, 1), (2, 2), (3, 3)} is the identity relation on A.
VARIOUS TYPES OF RELATIONS
Let A be a nonempty set. Then, a relation R on A is said to be
• (i) re exive if (a, a) ÎR for each a ÎA,
i.e., if a R a for each a ÎA.
• (ii) symmetric if (a, b) ÎR Þ (b, a) ÎR for all a, b ÎA,
i.e., if a R b Þ b R a for all a, b Î A.
(iii) transitive if(a, b) ÎR, (b, c) ÎR Þ (a, c) ÎR for all a, b, cÎA,
i.e., if a R b and b R c Þ a R c.
EQUIVALENCE RELATION A relation R in a set A is said to be an equivalence relation
if it is re exive, symmetric and transitive.
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SOLVED EXAMPLES
Let A be the set of all triangles in a plane and let R be a relation in A, de ned by R = {(C1 , C 2) : C1 @ C 2}.
Show that R is an equivalence relation in A.
The given relation satis es the following properties: (i) Re exivity
Let a be an arbitrary triangle in A. Then,
C @ C Þ (C, C) Î R for all values of C in A.
\ R is re exive.
(ii) Symmetry
LetC1, C2 ÎA such that(C1, C2) ÎR. Then,
(C1,C2)ÎR Þ C1 @C2 Þ C2 @C1
Þ (C2,C1)ÎR. \ R is symmetric.
(iii) Transitivity
Let C1, C2, C3 ÎA such that (C1, C2) ÎR and (C2, C3) ÎR. Then, (C1 , C2) Î R and (C2 , C3) Î R
Þ C1 @C2 andC2 @C3 Þ C1 @C2
Þ (C1,C3)ÎR.
\ R is transitive.
Thus, R is re exive, symmetric and transitive. Hence, R is an equivalence relation.
Let A be the set of all lines in xy-plane and let R be a relation in A, de ned by
R = {( L1 , L 2 ) : L1 | | L 2} .
Show that R is an equivalence relation in A.
Find the set of all lines related to the line y = 3x + 5.
The given relation satis es the following properties: (i) Re exivity
Let L be an arbitrary line in A. Then, L||L Þ (L,L)ÎR"LÎA.
Thus, R is re exive. (ii) Symmetry
Let L1, L2 ÎA such that(L1, L2) ÎR.Then, (L1,L2)ÎR Þ L1||L2
Þ L2||L1
Þ (L2,L1)ÎR. \ R is symmetric.
(iii) Transitivity
Let L1, L2, L3 ÎA such that (L1, L2) ÎR and (L2, L3) ÎR. Then,(L1, L2) ÎR and(L2, L3) ÎR
Relations 3
Þ L1||L2andL2||L3
Þ L1||L3
Þ (L1,L3)ÎR. \ R is transitive.
Thus R is re exive, symmetric and transitive.
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Hence, R is an equivalence relation.
The family of lines parallel to the line y = 3x + 5 is given by
y = 3x + k, where k is real.
