SOLUTION: Let A = {0, 1, 2, 3, 4, 5} and ∼ be a relation on A defined by x ∼ y if and only if x + y < 4. Is the relation an equivalence relation?

Question 1195955: Let A = {0, 1, 2, 3, 4, 5} and ∼ be a relation on A defined by x ∼ y if
and only if x + y < 4. Is the relation an equivalence relation?
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

Here's an article talking about Equivalence Relations
https://www.cuemath.com/algebra/equivalence-relations/

Focus on the subsection titled "Equivalence Relation Definition" and the three properties of Reflexive, Symmetric, and Transitive.
All three properties must hold if we want an equivalence relation.

The operator is reflexive if we can say a ~ a for all 'a' in the set A = {0, 1, 2, 3, 4, 5}
This works for 0 and 1 since
0+0 < 4 is true
1+1 < 4 is also true

But something like 2 doesn't work
2+2 < 4 is false
Therefore, 2 ~ 2 is a false statement.
Similar problems occur with values 3, 4 and 5.
The condition of Reflexivity doesn't hold true for all values in the set A.

Answer: No, it is not an equivalence relation.

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