Determining Elements in the Relation r for Set {0, 2, 4, 8}

In this article, we will discuss how to determine the elements in the relation r defined on the set A {0, 2, 4, 8} such that xry if and only if y x^2 / 2. This exploration will involve understanding the concept of the domain, function, and subset in relation to set theory.

Understanding the Relation r

Let us first define the relation r on a set A. The relation r is a subset of the Cartesian product of A with itself, A x A. This means that for any two elements x and y in A, xry if and only if the condition y x^2 / 2 is satisfied.

Applying the Condition to the Set {0, 2, 4, 8}

Given the set A {0, 2, 4, 8}, we will now apply the given condition to determine the elements in the relation r.

When x 0: y 0^2 / 2 0/2 0. Since 0 is in the set A, we include the pair (0, 0) in the relation r.

When x 2: y 2^2 / 2 4/2 2. Since 2 is in the set A, we include the pair (2, 2) in the relation r.

When x 4: y 4^2 / 2 16/2 8. Since 8 is in the set A, we include the pair (4, 8) in the relation r.

When x 8: y 8^2 / 2 64/2 32. Since 32 is not in the set A, we do not include the pair (8, 32) in the relation r.

Conclusion

After applying the given conditions, we find that the elements in the relation r for the set A are (0, 0), (2, 2), and (4, 8). This shows the importance of carefully checking the conditions and ensuring that all elements are within the given set.

To summarize, the relation r on the set A {0, 2, 4, 8} consists of the pairs (0, 0), (2, 2), and (4, 8). This exemplifies the application of set theory concepts and how they can be used to determine elements in a relation.

Understanding these concepts is fundamental to more advanced topics in mathematics and computer science, such as functions, graphs, and algorithms.