Question 9965 – Digital-Logic-Design
January 2, 2024UGC NET CS 2005 Dec-Paper-2
January 2, 2024ISRO-2017 December
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Question 1
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{n,1}
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{n, n}
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{n2, 1}
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{1, n2}
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S={1,2,3,4}
→ If a set is said to be equivalence, then the set must be
i) Reflexive
ii) Symmetric
iii) Transitive
i) Reflexive Relation: A relation ‘R’ on a set ‘A’ is said to be reflexive if (xRx)∀x∈A.
A = {1,2,3}
R = {(1,1), (2,2), (3,3)}
R = {(1,1), (2,2)} It is false.
R = {(1,1), (2,2), (3,3), (1,2)}
ii) Symmetric Relation: A relation on a set A is said to be symmetric if (xRy). Then (yRx)∀x,y∈A i.e., if ordered pair (x,y)∈R. Then (y,x)∈R ∀x,y∈A.
A={1,2,3}
R1={(1,2), (2,1)}
R2={(1,1), (2,2), (3,3), (1,2), (2,1), (2,3), (3,2)}
Transitive Relation:
A relation ‘R’ on a set ‘A’ is said to be transitive if (xRy) and (yRz), then (xRz)∀(x,y,z∈A).
A={1,2,3}
R1={ }
R2={(1,1)}
R3={(1,2), (3,1)}
R4={(1,2), (2,1), (1,1)}
⇾ A={1,2,3,4}
Largest ordered set is
S×S={(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4),(3,1),(3,2),(3,3),(3,4),(4,1),(4,2),(4,3),(4,4)}
⇒ Total = 16 = 42 = n2
Smallest ordered set = {(1,1),(2,2),(3,3),(4,4)}
⇒ Total=4=n
Note: In question, they are clearly mentioned that Rank of an Equivalence relation is equal to the number of induced Equivalence classes. Since we have maximum number of ordered pairs(which are reflexive, symmetric and transitive ) in largest Equivalence relation, its rank is always 1.
S={1,2,3,4}
→ If a set is said to be equivalence, then the set must be
i) Reflexive
ii) Symmetric
iii) Transitive
i) Reflexive Relation: A relation ‘R’ on a set ‘A’ is said to be reflexive if (xRx)∀x∈A.
A = {1,2,3}
R = {(1,1), (2,2), (3,3)}
R = {(1,1), (2,2)} It is false.
R = {(1,1), (2,2), (3,3), (1,2)}
ii) Symmetric Relation: A relation on a set A is said to be symmetric if (xRy). Then (yRx)∀x,y∈A i.e., if ordered pair (x,y)∈R. Then (y,x)∈R ∀x,y∈A.
A={1,2,3}
R1={(1,2), (2,1)}
R2={(1,1), (2,2), (3,3), (1,2), (2,1), (2,3), (3,2)}
Transitive Relation:
A relation ‘R’ on a set ‘A’ is said to be transitive if (xRy) and (yRz), then (xRz)∀(x,y,z∈A).
A={1,2,3}
R1={ }
R2={(1,1)}
R3={(1,2), (3,1)}
R4={(1,2), (2,1), (1,1)}
⇾ A={1,2,3,4}
Largest ordered set is
S×S={(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4),(3,1),(3,2),(3,3),(3,4),(4,1),(4,2),(4,3),(4,4)}
⇒ Total = 16 = 42 = n2
Smallest ordered set = {(1,1),(2,2),(3,3),(4,4)}
⇒ Total=4=n
Note: In question, they are clearly mentioned that Rank of an Equivalence relation is equal to the number of induced Equivalence classes. Since we have maximum number of ordered pairs(which are reflexive, symmetric and transitive ) in largest Equivalence relation, its rank is always 1.