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I have a bit of a problem with this. Our teacher told us to give 3 examples of binary relations which are non-antisymmetric. From all the examples I gave they were all symmetric.

From my knowledge, all asymmetric relation are also antisymmetric. So there is no example which is an asymmetric relation, and I have no idea of any other relation other than symmetric which is non-asymmetric and non-antisymmetric.

So my question is, do all non-antisymmetric relations have to be symmetric?

Edit:

antisymmetric: if for all $(a,b) \in R$ $\land (b,a) \in R \Rightarrow (a=b)$

asymmetric: non symmetric. if for all $(a,b) \in R \Rightarrow (b,a) \notin R$,

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    Please define "asymmetric" and "antisymmetric". – 2017-01-11
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    asymmetric means that if $(a, b) \in R$, then $(b,a) \notin R$. Antisymmetric means that if ($(a, b) \in R$ and $(b, a) \in R$), then $(a=b)$ – 2017-01-11
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    antisymmetric: if for all $(a,b) \in R$ $\land (b,a) \in R \Rightarrow (a=b)$, asymmetric: non symmetric. if for all $(a,b) \in R \Rightarrow (b,a) \notin R$, – 2017-01-11
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    since there already are so many answers i turn my answer into a comment:since you did not specify, i assume the following: symmetric: $a\sim b \leftrightarrow b\sim a$ antisymmetric: $a\sim b \leftrightarrow \neg b\sim a$ now for instance take the set $\Omega:=\left\{0,1\right\}$ and define: $0\sim 1, 1\sim 1, 0\nsim 0, 1\nsim 0$. Clearly, $\sim$ is neither symmetric, nor antisymmetric – 2017-01-11

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no, for example the relation on $\{a,b,c\}$ given by $\{(a,b),(b,c),(c,b)\}$ is not symmetric as $(a,b)$ appears and $(b,a)$ does not. It is also not antisymmetric as we have $(b,c)$ and $(c,b)$.

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    Great minds think alike. – 2017-01-11
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    does the same happen for poor ones? – 2017-01-11
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Go back to the definition. A symmetric relation has $xRy \Leftrightarrow yRx$ for all $x,y$. An antisymmetric relation has $(xRy \wedge yRx) \implies x=y$. Take the relation on $\{1,2,3,4\}$ consisting of $\{(1,2),(3,4),(4,3)\}$. Having $(1,2)$ and not $(2,1)$ makes it not symmetric. The other two pairs make it not antisymmetric.