My Discrete Mathematics textbook says the following :
A relation is symmetric/antisymmetric/transitive even if there’s one pair/triplet that satisfies the condition.
This probably means that if I have a relation that consists of a number of ordered pairs and suppose that only one pair satisfies the relation for symmetry , we can say that the relation is symmetric.
But then again , it makes the following statement:
Also a relation is not asymmetric/antisymmetric/transitive/asymetric if there’s one pair that does not satisfy the condition.
So I would interpret this as ‘even if there’s just one pair in a relation that doesn’t satisfy the condition for symmetry , the relation would not be considered as symmetric.
However don’t these two statements actually counter each other ? Suppose we have a relation where some pairs satisfy the condition for symmetry(say) and others don’t , then according to the first statement the relation would be symmetric since there’s atleast some pairs that satisfy the condition while according to the second , it would not be symmetric since there’s atleast one pair that doesn’t.
So what would be the correct interpretation?
Please help me out , I’m a beginner with relations.
Best Answer
I suspect that the word not is missing in the first statement immediately before the list of properties: as it stands, the statement is false. But even with that change the statement is very poorly worded. I suspect that the author is trying to say that finding one pair that satisfies the symmetry condition, for instance, is not enough to show that a relation is symmetric: all pairs must satisfy it in order for the relation to be symmetric. Similarly, finding one triplet satisfying the condition for transitivity does not show that a relation is transitive: you have to show that the condition is satisfied by all triplets.
The other statement is poorly worded but correct: if there is even just one pair (or triplet, in the case of transitivity) that does not satisfy the defining condition for the property, then the relation does not have the property.