Question

Let \(A=\\{1,2,3,4,5,6\\} .\) Observe that \(\varnothing \subseteq A \times A,\) so \(R=\varnothing\) is a relation on \(A .\) Draw a diagram for this relation.

Short answer

The diagram would be a set of 6 distinct points with no lines or arrows connecting them, representing the elements of 'A' with no relations as 'R' is an empty set.

Step-by-step solution

Understand the Binary Relation R

A relationship 'R' on a set 'A' is a collection of ordered pairs with both components in 'A'. Specifically, for this exercise, 'R' is defined as the empty set. It means that there are no ordered pairs in the relation.

Creating ordered pairs

Here, since 'R' is an empty set, it means that we do not have any ordered pairs. Normally, ordered pairs are represented as \((a, b)\) where 'a' and 'b' are elements of the set 'A'. But in our case 'R' does not contain any such pairs.

Draw the Diagram

A diagram for this would be a set of 6 points (representing the elements of the set A), with no lines or arrows between them (because there are no relations, as 'R' is empty).