UGC NET CS 2018 July - II Question 90

Question 90

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Which of the following statements is true ?

1. (1) (Z, ≤ ) is not totally ordered
2. (2) The set inclusion relation ⊆ is a partial ordering on the power set of a set S
3. (3) (Z, ≠ ) is a poset
4. (4) The directed graph is not a partial order
   

   

   Diagram for option (4) The directed graph is not a partial order

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

Explanation Question 90

Options (1) - FALSE
(Z, ≤ ) is not totally ordered
Z = (..., -3, -2, -1, 0, 1, 2, 3, ...}
Show that “less than or equal” relation
is a partial ordering on the set of integers.
– a<=a for every integer a (reflexive)
– a<=b, b<=a, then a=b (anti-symmetric)
– a<=b, b<=c, then a<=c (transitive)
Thus <= is a partial ordering on the set of
integers
(Z,<=) is a poset.

To prove that Z is Totally Ordered Set:
If (S,R) is a poset and every two
elements of S are comparable, S is called
a totally ordered set or linearly ordered
set.

Comparable: The elements a and b of a poset (Z, ≤ )
are comparable, if either aRb or bRa holds.

Option 2: - TRUE
The set inclusion relation ⊆ is a partial ordering on the power set of a set S (TRUE)
let set S = {a,b}
P(S) = {{a} , {b} ,{a,b} , Φ}

Show that  relation (S,⊆) is a partial ordering on the set of integers.
Take any two element from S, lets say s1 and s2
– s1⊆s1 for every element s1 (reflexive)
– s1⊆s2, s2⊆s1, then s1==s2 (anti-symmetric)
– s1⊆s2, s2⊆s3, then s1⊆s3 (transitive)
Thus ⊆ is a partial ordering on the power set of a set S
(S,⊆) is a poset.

Options 3: - FALSE
(Z,≠) is a poset?
Show that (Z,≠) is a partial ordering on the set of integers.
Take any two integers, then
– aRa -> a≠a for every integer a (not reflexive)
Reflexive Relation doesn't satisfy.
So, (Z,≠) is not a poset.

Options 4: - FALSE
Relation set for the directed graph is D={(a,a),(a,b),(b,b)}
Reflecxive,transitive property holds for the elements in the set.
Anti-symattric -> if R(a, b) with a ≠ b, then R(b, a) must not hold, so, Anti-symattric also holds.

References:
Partial Orderings - http://cse.iitkgp.ac.in/~debdeep/courses_iitkgp/FOC/slidespdf/POS.pdf
Partially ordered set - https://en.wikipedia.org/wiki/Partially_ordered_set
Total order - https://en.wikipedia.org/wiki/Total_order

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