Question

Determine whether the given relation is reflexive, symmetric, transitive, or none of these. Justify your answers. Let A be the set of all strings of 0’s, 1’s and 2’s of length 4. Define a relation R on A as follows: For all

s,t∈A,sRt⇔

the sum of the characters in s equals the sum of the characters in t.

Answer 1

The relation R is reflexive, symmetric, and transitive.

Step-by-step explanation:

The given relation R on set A is defined as follows: For all s, t ∈ A, sRt if and only if the sum of the characters in s equals the sum of the characters in t.

In order to determine whether the relation R is reflexive, symmetric, or transitive, let's analyze each property:

1. Reflexive: A relation is reflexive if every element in A is related to itself. In this case, for every string s ∈ A, the sum of characters in s will always be equal to the sum of characters in itself. Therefore, the relation R is reflexive.
2. Symmetric: A relation is symmetric if whenever s is related to t, then t is also related to s. In this case, if the sum of characters in s is equal to the sum of characters in t, then the sum of characters in t will also be equal to the sum of characters in s. Therefore, the relation R is symmetric.
3. Transitive: A relation is transitive if whenever s is related to t and t is related to u, then s is related to u. In this case, if the sum of characters in s is equal to the sum of characters in t, and the sum of characters in t is equal to the sum of characters in u, then the sum of characters in s will also be equal to the sum of characters in u. Therefore, the relation R is transitive.

Therefore, the relation R is reflexive, symmetric, and transitive.