2
$\begingroup$

I want to know why $$\{(a,b) \mid a\leq b\}$$ is not symmetric, when $$\{(a,b) \mid a\leq b\} =\{(a,b)| a<b \text{ or }a=b\}$$ So if a=b that means aRb and bRa so it is symmetric, right ?

Thanks all

$\endgroup$
  • 1
    $\begingroup$ Maybe you're confusing symmetric with reflexive? A relation $R$ is symmetric if $x R y \rightarrow y R x$ (for all $x, y$) - that is not the case for $\leq$. A relation is reflexive if $x R x$ (for all $x$) - that is true for $\leq$. $\endgroup$ – Magdiragdag Dec 13 '13 at 13:16
  • 1
    $\begingroup$ Since $1R2$, symmetric would mean $2R1$. $\endgroup$ – Thomas Andrews Dec 13 '13 at 13:20
  • $\begingroup$ @madiragdag thank you, I'm confused about that $\endgroup$ – user32104 Dec 13 '13 at 13:24
3
$\begingroup$

A relation is symmetric when for any values $a$ and $b$, if $a$ is related to $b$, then also $b$ is related to $a$. But in your case, for example, $3 \le 4$, but $4\not\le 3$.

$\endgroup$
  • $\begingroup$ thank you, but a<b OR a=b,so one of them should be true. (1,1) and (1,1) are symmetric right? $\endgroup$ – user32104 Dec 13 '13 at 13:20
  • 3
    $\begingroup$ The statement $a<=b$ if and only if $b<=a$ must be true for every pair of values $a$ and $b$. Showing that it's true for the pair $(1,1)$ does not help. $\endgroup$ – rogerl Dec 13 '13 at 13:22
1
$\begingroup$

HINT $(4, 7)$ means $(7, 4)$ should both be in $R$ if $R$ was symmetric.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.