Let = {1,2,3}and = {(, ): , |
2 −
2
| ≤ 5}. Write R as a set of ordered pair.
Mention whether R is reflexive, symmetric and transitive.
Answers
Answer:
Given A={1,2,3}
R={(a,b):a,b∈Aand
∣
∣
∣
a
2
−b
2
∣
∣
∣
≤5}
1) When a=1 and b=1,
∣
∣
∣
a
2
−b
2
∣
∣
∣
=
∣
∣
∣
∣
(1)
2
−(1)
2
∣
∣
∣
∣
∣
∣
∣
a
2
−b
2
∣
∣
∣
=0≤5
Thus, (1,1) is ordered pair of R
2) When a=1, b=2
∣
∣
∣
a
2
−b
2
∣
∣
∣
=
∣
∣
∣
∣
(1)
2
−(2)
2
∣
∣
∣
∣
∴
∣
∣
∣
a
2
−b
2
∣
∣
∣
=∣1−4∣
∴
∣
∣
∣
a
2
−b
2
∣
∣
∣
=∣−3∣=3≤5
Thus, (1,2) is ordered pair of R
3) When a=1, b=3,
∣
∣
∣
a
2
−b
2
∣
∣
∣
=
∣
∣
∣
∣
(1)
2
−(3)
2
∣
∣
∣
∣
∣
∣
∣
a
2
−b
2
∣
∣
∣
=∣1−9∣=∣−8∣
∴
∣
∣
∣
a
2
−b
2
∣
∣
∣
=8>5
This is not an ordered pair.
4) When a=2, b=1
∣
∣
∣
a
2
−b
2
∣
∣
∣
=
∣
∣
∣
∣
(2)
2
−(1)
2
∣
∣
∣
∣
∴
∣
∣
∣
a
2
−b
2
∣
∣
∣
=∣4−1∣=∣3∣
∴
∣
∣
∣
a
2
−b
2
∣
∣
∣
=3<5
Thus, (2,1) is an ordered pair.
5) When a=2, b=2
∣
∣
∣
a
2
−b
2
∣
∣
∣
=
∣
∣
∣
∣
(2)
2
−(2)
2
∣
∣
∣
∣
∣
∣
∣
a
2
−b
2
∣
∣
∣
=0≤5
Thus, (2,2) is ordered pair of R
6) When a=2, b=3
∣
∣
∣
a
2
−b
2
∣
∣
∣
=
∣
∣
∣
∣
(2)
2
−(3)
2
∣
∣
∣
∣
∴
∣
∣
∣
a
2
−b
2
∣
∣
∣
=∣4−9∣=∣−5∣
∴
∣
∣
∣
a
2
−b
2
∣
∣
∣
=5=5
Thus, (2,3) is an ordered pair.
7) When a=3, b=1
∣
∣
∣
a
2
−b
2
∣
∣
∣
=
∣
∣
∣
∣
(3)
2
−(1)
2
∣
∣
∣
∣
∴
∣
∣
∣
a
2
−b
2
∣
∣
∣
=∣9−1∣=∣8∣
∴
∣
∣
∣
a
2
−b
2
∣
∣
∣
=8>5
This is not an ordered pair.
8) When a=3, b=2
∣
∣
∣
a
2
−b
2
∣
∣
∣
=
∣
∣
∣
∣
(3)
2
−(2)
2
∣
∣
∣
∣
∴
∣
∣
∣
a
2
−b
2
∣
∣
∣
=∣9−4∣=∣5∣
∴
∣
∣
∣
a
2
−b
2
∣
∣
∣
=5=5
Thus, (3,2) is an ordered pair.
9) When a=3, b=3
∣
∣
∣
a
2
−b
2
∣
∣
∣
=
∣
∣
∣
∣
(3)
2
−(3)
2
∣
∣
∣
∣
∣
∣
∣
a
2
−b
2
∣
∣
∣
=0≤5
Thus, (3,3) is ordered pair of R
∴R={(1,1),(1,2),(2,1),(2,2),(2,3),(3,2),(3,3)}
R is reflexive as (a,b)∈R for every (a,b)∈A
R is symmetric as (a,b)∈R and (b,a)∈R
Relation is not transitive.
Answer:
Hey mate
ans- R={(1,1),(2,2),(3,3),(2,1),(1,2),(3,2),(2,3)}
R is refexive and symmetric but not transitive.
FIND STEP WISE ANSWER IN ATTACHMENT
Step-by-step explanation:
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