Suppose R is a partial order on A and S is a partial order o

Suppose R is a partial order on A and S is a partial order on B. Define a relation T on A × B such that (a1, b1) T (a2, b2) iff a1 R a2 and b1 S b2. Is T a partial order on A x B?

Solution

yes it is a partial order

a) reflexive

T is reflexive since

(a1,b1) T(a1, b1)

because a1 R a1 , R is reflexive

and b1 S b1 , S is reflexive

b) antisymmetric

if (a1,b1) T (a2,b2) and (a2,b2) T (a1,b1)

then

a1 R a2 and a2 R a1 so a1 = a 2 since R is antisymmetric

b1 S b2 and b2 S b1 so b1 = b2 since S is antisymmetric

So (a1,b1) = (a2,b2)

T is antisymmetric

c) transitive

(a1,b1) T (a2,b2) and (a2,b2) T (a3,b3)

then

a1 R a2 and a2 R a3 so a1 R a3 since R is transitive

b1 S b2 and b2 S b3 so b1 S b3 since S is transitive

so (a1,b1) T (a3,b3)

and T is transitive