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**Hartlw** R can’t be defined without ϵ. R=ϵ would be a circular definition, so ϵ has to be independent of R, ie, ϵ can’t be a relationship..

This is how I got there in case anyone is interested (it’s educational):

If A and B are sets, a Relation from A to B, is a subset of AXB.

Examples:

1) A={x,y,z}, B={1,2,3}, AXB={(x,1), (x,2), (x,3), (y,1)……..(z,3)}.

R ={(x,1), (x,2), (z,2), (z,3)}, and we say xR1,xR2,zR2,and zR3.

2) A={1,2}, B={1,2}, AXB={(1,1), (1,2), (2,1), (2,2)}

R ={(1,1), (2,2)}: Equal Relationship: 1=1 and 2=2.

R = {(1,2)}: Less Than relationship: 1<2.

R = {(2,1)}: Greater Than relationship: 2>1.

xϵA means x is a member of A.

Now suppose x is any member of A and y is any member of B, then a relation could be expressed as xRy which looks like xϵy in its most general form.

So could xϵy in its most general form be a relationship? No, because you need xϵy to define a relationship.