1. The following are relations on the set {1,2,3,4}. a.) { (2,1), (1,4), (2,4),
ID: 3708410 • Letter: 1
Question
1.
The following are relations on the set {1,2,3,4}.
a.) { (2,1), (1,4), (2,4), (4,3), (1,3), (4,4), (1,1), (4,1), (2,3) }
b.) { (1,1), (1,2), (2,2), (3,1), (3,3), (4,4), (1,3), (2,4) }
c.) { (1,3), (2,1), (3,2), (1,4) }
d.) { (1,3), (3,1), (2,3), (1,4), (3,2), (4,1) }
1. Which of the above is reflexive ?
2.) Which of the above is symmetric?
3.) Which of the above is antisymmetric?
4.) Which of the above is transitive?
2.
Write in the space provided the property or properties that each relation possesses (reflexive, symmetric, transitive). For example, write r,s if the relation is reflexive and symmetric.
1.) R is a relation on the set of all people. (a,b) is in R if a is taller than b.
2.) R is a relation on the set of integers > 0. (a,b) is in R if a divides b. (eg. 3 divides 3 and 3 divides 15 etc.)
3.) R is a relation on the set of all people. (a,b) is in R if a is married to b.
4.) R is a relation on the set of all nonnegative integers. (a,b) is in R if a and b have the same remainder when divided by 5. (eg. (7,12) is in R)
Explanation / Answer
Question 1
a) (2,2) is not a part of the relation. Hence, it is not reflexive.
(2,1) is a part of the relation, but (1,2) is not a part of the relation. Hence. it is not symmetric.
(1,4) and (4,1) are both a part of the relation. Hence, it is not antisymmetric.
For all (a.b) and (b,c) belonging to the relation , (a,c) belongs to the relation. Hence, it is transitive.
b) The relation is reflexive. For all a belonging to the set, (a,a) belongs to the relation. Hence, the relation is reflexive.
(1,2) belongs to the set, but (2,1) does not. Hence, it is not symmetric.
(1,3) and (3,1) both belong to the relation. Hence, it is not anti-symmetric.
(1,2) and (2,4) belong to the relation.But (1,4) does not.Hence, it is not transitive.
c)
(1,1) does not belong to the set.The relation is not reflexive.
(1,3) belongs to the set.But, (3,1) does not belong to the set. Hence, it is not symmetric.
If (a,b) belongs to the relation. And if (b,a) also belongs to the set, then a=b. Then, it is anti-symmetric. This condition holds true for the relation.
(1,3) and (3,2) belong to the relation.But (1,2) does not.Hence, it is not transitive.
d) (1,1) does not belong to the set. Hence it is not reflexive.
if (a,b) belongs to the set then (b,a) also belongs to the set. This holds true for the above relation. Hence, it is symmetric.
Both (1,3) and (3,1) belong to the set. Hence, it is not anti-symmetric.
(1,3) and (3,2) belong to the relation.But (1,2) does not.Hence, it is not transitive.
Thus,
1) Reflexive : b)
2) Symmetric: d)
3) Anti-symmetric: c)
4) Transitive: a)
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