State whether the binary relation is reflexive symmetric or

State whether the binary relation is reflexive, symmetric, or transitive.

(a) <

(b)

(c) {x,y | xmod3 = y mod3}

(d) {x,y | |x y| < 1}

(e) {x,y | |x y| 1}

(f) {x,y | |x y| > 1}

(g) {x,y | |x y| = 1}

Solution

(a)   a<a is not true so it is not reflexive

If a<b is true b<a can never be true so it is not symmetric

if a<b and b<c is true then a<c will always be true so it is transitive

(b)  a<=a(a<a or a=a) is true so it is reflexive

If a<=b is true then b<=a will always be true so it is symmetric

If a<=b and b<=c are true then always a<=c will be true so it is transitive

(c) For (a,a) a mod3=b mod3 will be true so it is reflexive

If (a,b) belongs to the given set (i.e. a mod3=b mod3) then (b,a) has to be in the set as b mod3=a mod3 so it is symmetric

If (a,b) and (b,c) belong to given set (i.e. a mod3=b mod3 and b mod3=c mod3) from these two it is clear that (a,c) is also in the set so it is transitive.

(d) (a,a) is in set as |a-a|=0<1 so it is reflexive

If (a,b) is in the set then for (b,a) |b-a|=|a-b|<1 therefore it is symmetric

If (a,b) and (b,c) belongs to the set then (a,c) may not belong to set consider example where a=0.5 b=0.2 and c=-0.7 then (a,b) and (b,c) belong to set but (a,c) [0.5-(-0.7)=1.2>1] does not belong to set so it is not transitive.