Which of the following statements is true?

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UGC NET Computer Science (Paper 2) July 2018 Paper
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  1. (Z, ≤ ) is not totally ordered
  2. The set inclusion relation ⊆ is a partial ordering on the power set of a set S
  3. (Z, ≠ ) is a poset
  4. The directed graph F1 R.S Madhu 31.03.20 D 6 is not a partial order

Answer (Detailed Solution Below)

Option 2 : The set inclusion relation ⊆ is a partial ordering on the power set of a set S
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Detailed Solution

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Option 1: (Z, ≤) is not totally ordered

This is false. As, ≤ is totally ordered. A partial ordered set with comparison is known as totally ordered set. Consider the set contains elements {1, 2, 3, 4}. This represents as totally ordered set. As 1<= 2, 2<=3, 3< = 4.

Option 2: The set inclusion relation ⊆ is a partial ordering on the power set of a set S

If we consider the set S = {a, b} . So possibilities with this set are {Ø, a, b , ab}. As, subset relation is reflexive, antisymmetric and transitive. So, it is partially ordered. Because if a is subset of b, then b is superset of a.

Option 3: (Z, ≠ ) is a poset

It is not partially ordered set. As, it follows symmetric property. Example if a is not equal to b , then b is also not equal to a. It can not be poset. Given statement is incorrect.

Option 4: The directed graph

F1 R.S Madhu 31.03.20 D 6

 is not a partial order

This statement is incorrect. This graph represents partial order. It can be represented as:

F1 R.S Madhu 06.05.20 D1

It is satisfying the property of partially ordered set. So, given directed graph is poset. It is reflexive, antisymmetric and transitive.

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