Question

The domain of a relation r is the set of integers. x is related to y under relation r if x²=y. Select the description that accurately describes relation r.

1) The relation r is a function.
2) The relation r is reflexive.
3) The relation r is symmetric.
4) The relation r is transitive.

Answer 1

The given relation is symmetric, which means if x is related to y, then y is also related to x.

Step-by-step explanation:

The given relation is defined as x² = y, where x and y are integers. The question asks for a description that accurately describes the relation.

1. The relation is a function: No. A function has a unique output for each input, but in this relation, the same output can have multiple inputs (e.g., both 2 and -2 have an output of 4).
2. The relation is reflexive: No. Reflexive means that every element is related to itself, but in this relation, it is not true for all integers.
3. The relation is symmetric: Yes. If x is related to y, then y is also related to x because if x² = y, then (-x)² = y as well.
4. The relation is transitive: No. Transitive means that if x is related to y and y is related to z, then x is related to z. However, in this relation, transitivity does not hold for all integers.

Based on the above analysis, option 3) The relation r is symmetric and accurately describes relation r.