Question

Consider the following statements:

Reflexive Property: Robot A is as rusty as itself.

Symmetric Property: If Robot A is as rusty as Robot B, then Robot B is as rusty as Robot A.

Transitive Property: If Robot A is as rusty as Robot B and Robot B is as rusty as Robot C, then Robot A is as rusty as Robot C.

A relation such as “is as rusty as” that is reflexive, symmetric, and transitive is an equivalence relation. Which of the following are equivalence relations?

a. Is rustier than

b. Has the same length as

c. Is opposite (for rays)

d. Is coplanar with (for lines)

Short answer

‘Has the same length as’ is an equivalence relation.

Step-by-step solution

Step 1. Reflexive property.

The reflexive property states that, if p is any number then $p=p$.

Step 2. Symmetric property.

Symmetric property states that, if $p=q$then $q=p$.

Step 3. Transitive property.

Transitive property states that, if $p=q$and $q=r$then $p=r$.

Step 4. Description of step.

A relation such as “Is as rusty as” that is reflexive, symmetric and transitive is an equivalence relation then “has the same length as” among the given options is an equivalence relation.

Therefore option ‘b’, that is, ‘has the same length as’ is a correct option.