- #1

Eclair_de_XII

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- 91

## Homework Statement

"Determine the max number of elements in a three-element set that is not reflexive, symmetric, or transitive?"

## Homework Equations

##a R b⇔(a,b)∈R##

## The Attempt at a Solution

Basically, my professor has stated that there are a total number of seven possible elements in a relation on a three element set that is not reflexive, symmetric,

*or*transitive.

I began again by listing all possible relations of a three element set: ##A=##{##x,y,z,##}. Let ##R## be a relation on ##A##. Then ##R=##{##(x,x),(x,y),(x,z),(y,x),(y,y),(y,z),(z,x),(z,y),(z,z)##}. There are nine elements in ##R## and by eliminating {##(x,x),(y,y),(z,z)##}, I already have fewer than the maximum number of elements in an ##R## that is not reflexive, symmetric, or transitive. And you kind of must nix those three elements for those three properties, anyway. I feel like I'm missing something... Can anyone point out what it is? Thanks.