A7P4: Recall that the set of integers is denoted by Z\'. Define a relation R on
ID: 3146519 • Letter: A
Question
A7P4: Recall that the set of integers is denoted by Z'. Define a relation R on the set Z by For all r,y Z, Ry if and only if 3| (x + 2,) Prove that R is an equivalence relation. (Make sure you understand the solution to Example 8.2.4 and the proofs of reflexivity and symmetry in Theorem 8.4.2 in the textbook before you start writing your solution to this problem. The algebra required to prove this relation is reflexive, symmetric, and transitive will be very similar, though not identical, to the algebra required in the solution to Example 8.4.2.)Explanation / Answer
x R y iff 3 | (x + 2y)
Check for reflexivity:
x + 2x = 3x
3 | 3x => 3 | (x + 2x)
=> x R x and R is reflexive.
Check for symmetry:
x ~ y => 3 | (x + 2y)
=> 3 | ((3x + 3y) - (x + 2y))
=> 3 | (2x + y)
=> y ~ x
Therefore, x is symmetric.
Check for transitivity:
Let x ~ y and y ~ z
=> 3 | (x + 2y) and 3 | (y + 2z)
Adding
=> 3 | ((x + 2y) + (y + 2z))
=> 3 | (x + 3y + 2z)
Since 3 | 3y
=> 3 | (x + 2z)
=> x ~ z.
Thus R is transitive.
Since R is reflexive, symmetric and transitive, it is an equivalence relation.