**To prove that R is an equivalence relation, we need to show that it satisfies three conditions: reflexivity, symmetry, and transitivity.**

- Reflexivity: For all strings a, we have (a, a) ∈ R. This is true because the length of a string is always equal to the length of itself.
- Symmetry: For all strings a and b, if (a, b) ∈ R, then (b, a) ∈ R. This is true because if the length of string a is equal to the length of string b, then the length of string b is also equal to the length of string a.
- Transitivity: For all strings a, b, and c, if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R. This is true because if the length of string a is equal to the length of string b, and the length of string b is equal to the length of string c, then the length of string a is also equal to the length of string c.

Therefore, R is reflexive, symmetric, and transitive, and is therefore an equivalence relation.

*R*={(*a*,*b*)∣ length of string *a*=length of string *b*}