Question
Download Solution PDFWhich of the following statements is true?
- (Z, ≤ ) is not totally ordered
- The set inclusion relation ⊆ is a partial ordering on the power set of a set S
- (Z, ≠ ) is a poset
- The directed graph
is not a partial order
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFOption 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
is not a partial order
This statement is incorrect. This graph represents partial order. It can be represented as:
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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