I have this problem which I can't figure out the first part of:
Let $A=\{2,3,4,8,9,12\}$, and let the relation $R$ on $A$ be defined by $aRb$ if and only if $(a\mid b\wedge a\ne b)$.
I believe that the part $b\wedge a \ne b$ means that you cannot have a relation of the type $(b,b)$, or another way to say it is that the relation cannot be reflexive, but I am really not sure.
Any help is appreciated. Thanks.
The definition of the relation rules out any pair of the form $( a,a)$.
It means that the relaton is irreflexive.
But note that ruling out all pairs of this form , though a sufficient condition for R to be non-reflexive, is not necessary. Ruling out all such pairs is necessary for a relation to be irreflexive.
The absence of only one possible pair of this form is already a sufficient condition for R not to be reflexive.
I mean that if your relation contained $( 2,2 )$ , $( 3,3 )$ , $( 4,4 )$ , $( 8,8 )$ , $(9 ,9 )$ but not the pair$( 12, 12)$ , it would already be non- reflexive.