Question

show that the relation r on the set of all bit strings such that s r t if and only if s and t contain the same number of 1s is an equivalence relation. what is the equivalence class of the bit string 011 for r?

Answer 1

The relation r on the set of all bit strings is an equivalence relation, where s r t if and only if s and t contain the same number of 1s. The equivalence class of the bit string 011 for r is the set of bit strings that contain the same number of 1s as 011.

Step-by-step explanation:

To show that the relation r on the set of all bit strings is an equivalence relation, we need to prove three properties: reflexivity, symmetry, and transitivity.

1. Reflexivity: For any bit string s, s contains the same number of 1s as itself. Therefore, s r s holds.

2. Symmetry: If s r t, it means s and t contain the same number of 1s. But since the number of 1s is the same, t and s also contain the same number of 1s. Hence, t r s holds.

3. Transitivity: If s r t and t r u, it means s and t have the same number of 1s, and t and u have the same number of 1s. Since the number of 1s is the same for both pairs, we can conclude that s and u also have the same number of 1s. Therefore, s r u holds.

The equivalence class of the bit string 011 for the relation r would be the set of all bit strings that contain the same number of 1s as 011.