Let
R be a relation from N
to N
defined by R = {(a,
b):
a,
b
∈
N
and a
= b2}.
Are the following true?
(i) (a,
a)
∈
R, for all
a ∈
N (ii) (a,
b)
∈
R, implies (b,
a)
∈
R
(iii) (a,
b)
∈
R, (b,
c)
∈
R implies (a,
c)
∈
R.
Justify your answer
in each case.
Let R be a relation from N to N defined by R = {(a, b): a, b ∈ N and a = b2}. Are the following true?
(i) (a, a) ∈ R, for all a ∈ N (ii) (a, b) ∈ R, implies (b, a) ∈ R
(iii) (a, b) ∈ R, (b, c) ∈ R implies (a, c) ∈ R.
Justify your answer in each case.
R
= {(a,
b):
a,
b
∈
N
and a
= b2}
(i) It
can be seen that 2 ∈
N;however,
2 ≠
22
= 4.
Therefore,
the statement “(a,
a)
∈
R, for all
a ∈
N”
is not true.
(ii) It
can be seen that (9, 3) ∈
N because
9, 3 ∈
N and
9 = 32.
Now,
3 ≠
92
= 81; therefore, (3, 9) ∉
N
Therefore,
the statement “(a,
b)
∈
R, implies (b,
a)
∈
R” is not true.
(iii) It
can be seen that (9, 3) ∈
R, (16, 4) ∈
R because 9, 3, 16, 4 ∈
N and
9 = 32
and 16 = 42.
Now,
9 ≠
42
= 16; therefore, (9, 4) ∉
N
Therefore,
the statement “(a,
b)
∈
R, (b,
c)
∈
R implies (a,
c)
∈
R” is not true.
R = {(a, b): a, b ∈ N and a = b2}
(i) It can be seen that 2 ∈ N;however, 2 ≠ 22 = 4.
Therefore, the statement “(a, a) ∈ R, for all a ∈ N” is not true.
(ii) It can be seen that (9, 3) ∈ N because 9, 3 ∈ N and 9 = 32.
Now, 3 ≠ 92 = 81; therefore, (3, 9) ∉ N
Therefore, the statement “(a, b) ∈ R, implies (b, a) ∈ R” is not true.
(iii) It can be seen that (9, 3) ∈ R, (16, 4) ∈ R because 9, 3, 16, 4 ∈ N and 9 = 32 and 16 = 42.
Now, 9 ≠ 42 = 16; therefore, (9, 4) ∉ N
Therefore, the statement “(a, b) ∈ R, (b, c) ∈ R implies (a, c) ∈ R” is not true.
