Question

Which of the following relations is not a partial ordering on the set of positive integers?

		R1 = { (a,b) | a >= b }
		R2 = { (a,b) | a <= b }
		R3 = { (a,b) | a is a multiple of b }
		R4 = { (a,b) | a ≡ b (mod 8) }

Answer 1

A relation is called partially ordered set if it follows following property

• Reflexive
• Antisymmetric
• Transitive

1. Relation : a>=b

Here the relation is reflexive, transitive and antisymmetric all.

Therefore the relation is partially ordered set.

2.Relation : a <= b

Here the relation is reflexive since (x, x) for all values of x in R.

The relation is transitive.

It is also antisymmetric since if (x<=y) and (y<=x) iff x=y.

Therefore the relation is partially ordered set.

3. (a, b) : a is multiple of b.

Here the relation is reflexive since every number is multiple of itself. The relation is antisymmetric and transitive. Therefore the relation is partially ordered set.

4. a = b(mod 8)

For number greater than 8 , the relation is not reflexive. Since it is not reflexive, it is not partially ordered set.

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