Question

A relation ⪯ is defined on ℚ by

?⪯? if and only if ?=?+6? for some non-negative integer ?.

Prove that ⪯ is a partial order.

Answer 1

Reflexivity

Now, 0 is a non-negative integer.

Thus, is reflexive.

Antisymmetry

for some non-negative integers k and k'.

Now, since both k and k' are non-negative, the above equality is satisfied only if k = k' = 0.

which implies that a = b.

Thus, is antisymmetric.

Transitivity

for some non-negative integers k and k'.

where k'' = k + k' and k'' is a non-negative integer.

Thus, is transitive.

Since, is reflexive, antisymmetric and transitive, it implies that is a partial order.

Hence Proved

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