Question

Given an example of a relation. Which is (i) Symmetric but neither reflexive nor transitive. (ii) Transitive but neither reflexive nor symmetric. (iii) Reflexive and symmetric but not transitive. (iv) Reflexive and transitive but not symmetric. (v) Symmetric and transitive but not reflexive.

Answer 1

(i)

Let set

A={ 5,6,7 }.

Let R be a relation on A defined as R={ ( 5,6 ),( 6,5 ) }.

( 5,5 )∉R. Hence R is not reflexive.

( 5,6 )∈R,.and ( 6,5 )∈R. Hence, Ris symmetric.

( 5,6 ),( 6,5 )∈R, but ( 5,5 )∉R, hence Ris not transitive.

Hence, the relation R={ ( 5,6 ),( 6,5 ) } on set A={ 5,6,7 } is symmetric, but neither reflexive nor transitive.

(ii)

Let relation R in set R of real numbers be defined as R={( a,b ):a<b}.

( a,a )∉R, since for all values of a, a∈R, a=a. Hence, R is not reflexive.

Let ( a,b )∈R, thus a<b. Then ( b,a )∉R since b>a. Hence, Ris not symmetric.

Let ( a,b ),( b,c )∈R, thus a≤b and b≤c. So, a≤c implies ( a,c )∈R, hence Ris transitive.

Therefore, the given relation R={( a,b ):a<b} in set R of real numbers is transitive, but neither reflexive nor symmetric.

(iii)

Let A={ 4,6,8 }. Let a relation R on A be defined as R={ ( 4,4 ),( 6,6 ),( 8,8 ),( 4,6 ),( 6,4 ),( 6,8 ),( 8,6 ) }.

( a,a )∈R, for all values a∈A. Hence, R is reflexive.

( a,b )∈R, for all values of, a,b∈Aand ( b,a )∈R for all values of, a,b∈A. Hence, Ris symmetric.

( 4,6 ) and ( 6,8 )∈R, but ( 4,8 )∉R, hence Ris not transitive.

Therefore, the given relation R={ ( 4,4 ),( 6,6 ),( 8,8 ),( 4,6 ),( 6,4 ),( 6,8 ),( 8,6 ) } in set A={ 4,6,8 } is reflexive and symmetric but not transitive.

(iv)

Let a relation R in set R of real numbers be defined as R={ ( a,b ): a 3 ≥ b 3 }.

( a,a )∈R, since a 3 = a 3 for all values of a, a∈A. Hence, R is reflexive.

( 2,1 )∈R, since 2 3 > 1 3 or 8>1, but ( 1,2 )∉R since, 1 3 < 2 3 or 1<8. Hence, Ris not symmetric.

Let ( a,b ) and ( b,c )∈R. Hence a 3 > b 3 and b 3 > c 3 , so a 3 > c 3 . Thus ( a,c )∈R, hence Ris transitive.

Therefore, the given relation R={ ( a,b ): a 3 ≥ b 3 } in set R of real numbers is reflexive and transitive, but not symmetric.

(v)

Let A={ −5,−6 }. Let a relation R on A be defined as R={ ( −5,−6 ),( −6,−5 ),( −5,−5 ) }.

( −6,−6 )∉R. Hence R is reflexive.

( −5,−6 )∈R, and ( −6,−5 )∈R. Hence, Ris symmetric.

( −5,−6 ) and ( −6,−5 )∈Rimply ( −5,−5 )∈R, hence Ris transitive.

Therefore, the given relation R={ ( −5,−6 ),( −6,−5 ),( −5,−5 ) } on the set A={ −5,−6 } is symmetric and transitive, but not reflexive.