# Why $\{(a,b) | a\leq b\}$ is not symmetric?

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

• 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 $\leq$. – Magdiragdag Dec 13 '13 at 13:16
• Since $1R2$, symmetric would mean $2R1$. – Thomas Andrews Dec 13 '13 at 13:20
• @madiragdag thank you, I'm confused about that – user32104 Dec 13 '13 at 13:24

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 $4\not\le 3$.
• 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. – rogerl Dec 13 '13 at 13:22
HINT $(4, 7)$ means $(7, 4)$ should both be in $R$ if $R$ was symmetric.