A question from the back of my Discrete Math classes textbook. I cant think of any example where (a,b) belongs to R and (b,a) belongs to R but a $\neq$ b. The answer in the back literally just says "no". This is my first question on here please be gentle.
2026-03-27 03:44:14.1774583054
Let S be the set of nonzero integers. Define a relation R on S by letting aRb mean that b/a is an integer. Is R an antisymmetric relation on S?
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Welcome to MSE! The answer is ''depends''.
Define the relation $aRb$ iff $a$ divides $b$, i.e., there exists $c$ with $a\cdot c=b$.
If the set on which the relation is defined is the set of integers, then $-1$ divides $1$ and $1$ divides $-1$, but $1\ne -1$.
If the set on which the relation is defined is the set of natural numbers, then the relation is antisymmetric.