(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 (as 1 < 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) ∈R i.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): a3 ≥ b3}
Clearly (a, a) ∈ R as a3 = a3.
∴ R is reflexive.
Now,
(2, 1) ∈ R (as 23 ≥ 13)
But,
(1, 2) ∉ R (as 13 < 23)
∴ R is not symmetric.
Now,
Let (a, b), (b, c) ∈ R.
⇒ a3 ≥ b3 and b3 ≥ c3
⇒ a3 ≥ c3
⇒ (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.
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(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.
...
Step-by-step explanation:
• (i) Let A = {5, 6, 7}
•
•
• Define a relation R on A as R = {(5, 6), (6, 5)}
•
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Student Discussion
Properties of Relation is
A realtion R on set A is reflexive if aRa for all a belongs to A i.e. is (a,a) belongs to R for all a belongs to R => each element a of A is related to itself.
Ex: Let A = {a,b} and R = {(a,a),(a,b),(b,a)} then R is reflexive as aRa belongs to R but it is not reflexive for pair (b,b) does not belongs to R.
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