Let A = { a,b,c,d } Construct relations on A of the following types : i) not reflexive, not symmetric, transitive ii) reflexive, not symmetric, transitive iii) reflexive, symmetric, not transitive iv) reflexive, symmetric, transitive v) not reflexive, not symmetric, not transitive
Answers
Answer:
Answer
(i)
Let A={5,6,7}
Define a relation R on A as R={(5,6),(6,5)}
Relation R is not reflexive as (5,5),(6,6),(7,7)∈
/
R.
Now, as (5,6)∈R and also (6,5)∈R, R is symmetric.
⇒(5,6),(6,5)∈R, but (5,5)∈
/
R
∴R is not transitive.
Hence, relation R is symmetric but not reflexive or transitive.
(ii)
Consider a relation R in R defined as
R={(a,b):a<b}
For any a∈R, we have (a,a)∈
/
R since a cannot be strictly less than a itself. In fact, a=a.
∴R is not reflexive.
Now, (1,2)∈R(as1<2)
But, 2 is not less than 1.
∴(2,1)∈
/
R
∴R is not symmetric.
Now, let (a,b),(b,c)∈R.
⇒a<b and b<c
⇒a<c
⇒(a,c)∈R
∴R is transitive.
Hence, relation R is transitive but not reflexive and symmetric.
(iii)
Let A={4,6,8}
Define a relation R on A as:
A={(4,4),(6,6),(8,8),(4,6),(6,4),(6,8),(8,6)}
Relation R is reflexive since for every {a∈A,(a,a)∈Ri.e.,(4,4),(6,6),(8,8)}∈R
Relation R is symmetric since (a,b)∈R⇒(b,a)∈R for all a,b∈R.
Relation R is not transitive since (4,6),(6,8)∈R, but (4,8)∈
/
R.
Hence, relation R is reflexive and symmetric but not transitive.
(iv)
Define a relation R in R as:
R={(a,b):a
3
≥b
3
}
Clearly (a,a)∈R as a
3
=a
3
.
∴R is reflexive.
Now, (2,1)∈R (as2
3
≥1
3
)
But, (1,2)∈
/
R(as1
3
<2
3
)
∴R is not symmetric.
Let (a,b),(b,c)∈R
⇒a
3
≥b
3
and b
3
≥c
3
⇒a
3
≥c
3
⇒(a,c)∈R
∴R is transitive.
Hence, relation R is reflexive and transitive but not symmetric.
(v)
Let A={−5,−6}.
Define a relation R on A as:
R={(−5,−6),(−6,−5),(−5,−5)}
Relation R is not reflexive as (−6,−6)∈
/
R
Relation R is symmetric as (−5,−6)∈R and (−6,−5)∈R
It is seen that (−5,−6),(−6,−5)∈R.
Also, (−5,−5)∈R.
∴ the relation R is transitive.
Hence, relation R is symmetric and transitive but not reflexive.