Question

ز Let s = {1,2,3,4) construct a relation on s which is reflexive , not Symmetric & not transitive? ​

Answer 1

we define a relation R
1
​
as
R
1
​
={(1,1),(2,2),(3,3),(4,4),(1,2),(2,3),(1,3)}
Then it is easy to check that R
1
​
is reflexive, transitive but not symmetric. Students are advised to write other relations of this type.
(ii) Define R
2
​
as: R
2
​
={(1,2),(2,1)}
Ti is clear that R
2
​
is symmetric but neither reflexive nor transitive. Write other relations of this type.
(iii) We define r
3
​
as follows:
R
3
​
={(1,1),(2,2),(3,3),(4,4),(1,2),(2,1)}.
Then evidently R
3
​
is reflexive, symmetric and transitive, that is, R
3
​
is an equivalence relation on A.
(1, 2) ∈R
3
​
,(2,1)∈R
3
​
⇒(1,1)∈R
3
​

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Answer 2

Answer:

(i) we define a relation R1 as 

R1={(1,1),(2,2),(3,3),(4,4),(1,2),(2,3),(1,3)}

Then it is easy to check that R1 is reflexive, transitive but not symmetric. Students are advised to write other relations of this type.

(ii) Define R2 as: R2 ={(1,2),(2,1)}

Ti is clear that R2 is symmetric but neither reflexive nor transitive. Write other relations of this type.

(iii) We define r3 as follows: 

R3 ={(1,1),(2,2),(3,3),(4,4),(1,2),(2,1)}.

Then evidently R3 is reflexive, symmetric and transitive, that is, R3 is an equivalence relation on A.

(1, 2) ∈R3,(2,1)∈R3⇒(1,1)∈R3