Define a relation R from Z to Z by (x,y) are in the relation if and only if the absolute value of (x-y) is less than or equal to two. Is R symmetric?

To determine whether the relation R is symmetric, we need to determine if for every element (x, y) in R, the element (y, x) is also in R.

In this case, the relation R is defined as (x, y) are in the relation if and only if the absolute value of (x - y) is less than or equal to two.

To check if R is symmetric, we need to verify whether for any (x, y) in R, (y, x) is also in R.

Let's consider an example to better understand this relation. Let's say we have (x, y) = (3, 7).

In this case, the absolute value of (x - y) = |3 - 7| = 4, which is not less than or equal to two.

Since (3, 7) is not in R, we can conclude that (7, 3) does not need to be in R, as we don't have any elements (x, y) in R to consider.

To make a final conclusion, we need to check all possible elements (x, y) in R. However, since we already found an element (x, y) that is not in R, we can conclude that the relation R is not symmetric.

In summary, the relation R is not symmetric.