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It's not clear to me why this relation is not total:

$\{(x,y)|x=y\}$

The way I interpret this is, since x = y, it is always true that (x,y) or (y,x) is part of the relation. Doesn't this imply that the relation is total?

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    What is the definition of total? – 2017-01-01
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    Either (x,y) or (y,x) has to be part of the relation, for all vertices x and y – 2017-01-01

1 Answers 1

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This is not true. Suppose $x=2$ and $y=3$. Neither $(2,3)$ nor $(3,2)$ are members of the relation.

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    But x has to be equal to y in order for the tuple to be part of the relation, doesn't it? – 2017-01-01
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    Yes, that's correct. Looks like you're reasoning is circular. Your argument seems to be the following: If $(x,y)$ is in the relation, then $x=y$ which means that $(x,y)$ is in the relation. But you assume $(x,y)$ was in the relationship to begin with. – 2017-01-01