Are These Relations Symmetric?

In summary, out of the given relations, only the second one (x~y if and only if xy >= 0) is symmetric, as it is true for both x+y and y+x. The others may not hold true for both variables, thus making them not symmetric.
  • #1
Shark1
1
0
Determine which of these relations are symmetric

1) x~y if and only if x-y is positive
2) x~y if and only if xy >= 0
3) x~y if and only if x+2y is positive
4) x~y if and only if x+y is positive
5) x~y if and only if x+y is odd

I thought all but 1) but this was wrong.
The only one I am sure that is symmetric is 2)
 
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  • #2
It is certainly true that if x+ y is positive, so is y+ x.

It is certainly true that if x+ y is odd, so is y+ x.

What about "x+ 2y is positive"? Can you find x such that x+ 2y is positive but y+ 2x is not?
 

FAQ: Are These Relations Symmetric?

What does it mean for a relation to be symmetric?

A relation is said to be symmetric if for every pair of elements (a, b) in the relation, the reverse pair (b, a) is also in the relation.

How can I determine if a relation is symmetric?

To determine if a relation is symmetric, you can check if the reverse of each pair of elements in the relation is also in the relation. If this is true for all pairs, then the relation is symmetric.

What is an example of a symmetric relation?

An example of a symmetric relation is the "is equal to" relation, where if a = b, then b = a. Another example is the "is a sibling of" relation, where if person A is a sibling of person B, then person B is also a sibling of person A.

Can a relation be both symmetric and asymmetric?

No, a relation cannot be both symmetric and asymmetric. These are opposite properties - a relation is symmetric if the reverse of every pair is also in the relation, while a relation is asymmetric if the reverse of any pair is not in the relation.

Is the "less than" relation symmetric?

No, the "less than" relation is not symmetric. For example, if a < b, then it is not true that b < a. Therefore, the reverse of the pair (b, a) is not in the relation, making it asymmetric.

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