Answer
Reflexive
Symmetric
Transitive
Work Step by Step
A is the “absolute value” relation on R: For all real numbers
x and y,x A y ⇔ |x| = |y|.
Reflexive: for all x $\in$ Z xAx
true since |x| = |x| for every x
Symmetric: for all x,y $\in$ Z if xAy then yAx
xAy <=> |x| = |y| >> |y| = |x| (commutive law) = yAx (by def of A)
Transitive: for all x,y, and z $\in$ Z if (xAy and yAz) then xAz
xAy <=> |x| = |y| >> |y| = |x| commutive law--(1)
yAz <=> |y| = |z| -- (2)
**sub 1 in 2 |x| = |z|
xAz (By definition of A) [that's what we needed to show]
