Sets And Relation

Question 1
Consider the binary relation R = {(x, y), (x, z), (z, x), (z, y)} on the set {x, y, z}. Which one of the following is TRUE?
A
R is symmetric but NOT antisymmetric
B
R is NOT symmetric but antisymmetric
C
R is both symmetric and antisymmetric
D
R is neither symmetric nor antisymmetric
       Engineering Mathematics        Sets And Relation       2009
Question 1 Explanation: 
Symmetric Relation: A relation R on a set A is called symmetric if (b,a) € R holds when (a,b) € R.
Antisymmetric Relation: A relation R on a set A is called antisymmetric if (a,b)€ R and (b,a) € R then a = b is called antisymmetric.
In the given relation R, for (x,y) there is no (y,x). So, this is not Symmetric. (x,z) is in R also (z,x) is in R, but as per antisymmetric relation, x should be equal to z, where this fails.
So, R is neither Symmetric nor Antisymmetric.
Question 2
     
A
B
C
D
       Engineering Mathematics        Sets And Relation       Gate-2007
Question 2 Explanation: 
π4 = refines every partition. So it has to be bottom of poset diagram.
And, neither π2 refines π3, nor π3 refines π2.
Here, only π1 refined by every set, so it has to be at the top.
Finally, option C satisfies all the property.
Question 3
Let (S, ≤) be a partial order with two minimal elements a and b, and a maximum element c. Let P: S → {True, False} be a predicate defined on S. Suppose that P(a) = True, P(b) = False and P(x) ⇒ P(y) for all x, y ∈ S satisfying x≤y, where stands for logical implication. Which of the following statements CANNOT be true?    
A
P(x) = True for all x ∈ S such that x ≠ b
B
P(x) = False for all x ∈ S such that x ≠ a and x ≠ c
C
P(x) = False for all x ∈ S such that b ≤ x and x ≠ c
D
P(x) = False for all x ∈ S such that a ≤ x and b ≤ x
       Engineering Mathematics        Sets and Relation       Gate-2003
Question 3 Explanation: 
c is the maximum element.
a or b the minimal element in set.
P(a) = True for all x ∈ S such that a ≤ x and b ≤ x.
Option D is False.
There are 3 questions to complete.