Consider the relation R given by divisibility on positive integers that is xRy <-> x|y Is this relation reflexive? symmetric? anti-symmetric? transitive??
I understand it is reflexive and transitive but i got a little confuse between symmetric and anti-symmetric.
Because the solution i have for anti-symmetric mention is True given that x|y = 1 and y|x = 1, hence x = y having the same value. so i was wondering if i use the argument for anti-symmetric on symmetric, won't is make the whole thing symmetric too?
Working straight from the definition on wikipedia, is suffices to show that if $a|b$ and $b|a$, then $a=b$. Just expand these hypotheses out as $a=mb$ and $b=na$ for some $n,m\in\mathbb{N}$ and notice that both $n=m=1$. Assuming you are working with $\mathbb{N}$. This shows your relation is anti-symmetric.
Let's show it's not symmetric just because we can. Notice that $5|10$ but $10\not|5$, so it's not true in general that $aRb$ implies $bRa$ for each $a$ and $b$.