Question

Let R be a relation defined by

R={(a, b):ab+2>0}. Verify the following

i) (a, b ) belongs to R and (b, c) belongs to R implies that (a, c) belongs to R

Answer 1

aRb <=> 2+ab >0

a. is reflexive because a*a for any real number except 0 will be positive hence >0, and if a=0 then a*a + 2 >0.

b. if a*b + 2 > 0 then b*a + 2 will also be > 0, hence symmetric.

c. a=-2,b=0, -2*0 + 2>0, ab+2> 0

b=0, and if c=4 then 0*4 + 2 > 0

but a is not related to c, because a=-2, c=4, and -2*4 + 2 < 0

Hence, the given relation is reflexive and symmetric but not transitive.

So,) (a, b ) belongs to R and (b, c) belongs to R implies that (a, c) belongs to R

Is wrong