Question

Define the following relations on A={5,6,7,8}.

R₀​={(5,5),(6,6),(6,7),(8,6),(7,7),(8,8),(6,8)}
R₁​={(5,5),(6,7),(7,6),(6,6),(5,6),(7,5),(8,8),(5,7),(7,7)}
R₂={(5,6),(7,7),(8,8),(5,5),(6,6)}
R₃​={(5,6),(5,8),(6,8),(5,5),(6,6),(7,7),(8,8)}
R₄​={(7,7),(5,5),(6,6),(8,8)}​
Which relations are reflexive?

Answer 1

Upon examination, the reflexive relations on set A={5,6,7,8} are R₂ and R₄ as both include all pairs (a, a) for each element 'a' in set A.

Step-by-step explanation:

In mathematics, a reflexive relation on a set A is defined as a relation where every element is related to itself. This means that for a relation to be reflexive, for every element 'a' in set A, the pair (a, a) must be included in the relation.

Given the set A = {5,6,7,8}, let's examine the provided relations:

• R₀​={(5,5),(6,6),(6,7),(8,6),(7,7),(8,8),(6,8)} is not reflexive because it is missing the pair (7, 7).
• R₁​={(5,5),(6,7),(7,6),(6,6),(5,6),(7,5),(8,8),(5,7),(7,7)} is not reflexive because it is missing the pair (8, 8).
• R₂={(5,6),(7,7),(8,8),(5,5),(6,6)} is reflexive because it includes all pairs (a, a) for each element a in set A.
• R₃​={(5,6),(5,8),(6,8),(5,5),(6,6),(7,7),(8,8)} is not reflexive because it is missing the pair (7, 7).
• R₄​={(7,7),(5,5),(6,6),(8,8)} is reflexive because it includes all pairs (a, a) for each element a in set A.

So, the reflexive relations on set A are R₂ and R₄​.