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4. (10 pts) Determine whether the following binary relations are
reflexive,
symmetric, antisymmetric and transitive:
a. x R y ⇔ xy ≥ 0 ∀ x, y ϵ R Reflexive - Any relation to be reflexive,
(x,x) should belong to R.
If we consider any value of x then x*x will always be an positive
value >0. For
example X=2, Y=2 2*2 > = 0 or X= -4 Y= -4, -4*-4> = 0
therefore we can say R is
reflexive. Symmetric - any relation to be symmetric, (x,y) should
belong to R and (y,x)
should also belong to R. here for any value of x and y if (x,y) belongs
to R i.e,
x*y>=0 then y*x will also be > = 0 thus (y,x) will also belong
to R. It is also
symmetric. Not antisymmetric because it is symmetric. Transitive -
any relation to be transitive, must hold if (x,y) and (y,z) belongs to R
then (x,z) should belong to R. When x*y>=0 and y*z>=0 the
we can say x*z will
also be >=0, thus (x,z) belongs to R. Is∀an
relation.
x equivalence
,
x> 0 [ x ] = {∀ y∨ y> 0 } ,
x< 0 [ x ] = { ∀ y| y< 0 } ,