Set Theory & Algebra (original) (raw)

The time complexity of computing the transitive closure of a binary relation on a set of n elements is known to be

Given a boolean function f (x1, x2, ..., xn), which of the following equations is NOT true

• f(x1​, x2​, …, xn​) = x1'f(x1​, x2​, …, xn​) + x1​f(x1​, x2​, …, xn​)
• f(x1, x2, ..., xn) = x2f(x1, x2, …, xn) + x2'f(x1, x2, …, xn)
• f(x1, x2, ..., xn) = xn'f(x1, x2, …, 0) + xnf(x1, x2, …,1)
• f(x1, x2, ..., xn) = f(0, x2, …, xn) + f(1, x2, ..., xn)

A binary operation ⊕ on a set of integers is defined as x ⊕ y = x2 + y2. Which one of the following statements is TRUE about ⊕?

• Commutative but not associative
• Both commutative and associative
• Associative but not commutative
• Neither commutative nor associative

Consider the set S = {1, ω, ω2}, where ω and w2 are cube roots of unity. If * denotes the multiplication operation, the structure (S, *) forms

Which one of the following in NOT necessarily a property of a Group?

• Existence of inverse for every element

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?

• R is symmetric but NOT antisymmetric
• R is NOT symmetric but antisymmetric
• R is both symmetric and antisymmetric
• R is neither symmetric nor antisymmetric

For the composition table of a cyclic group shown below

Which one of the following choices is correct?

The number of students in three classes is in the ratio 3:13:6. If 18 students are added to each class, the ratio changes to 15:35:21. The total number of students in all the three classes in the beginning was:
**[GATE 2021 || Set-2 || 2 Marks || MCQ]

If P, Q, R are subsets of the universal set U, then the value of following expression is

(P ∩ Q **∩ R)∪ (Pc **∩ Q **∩ R) ∪ Qc ∪ Rc

Let S be a set of n elements. The number of ordered pairs in the largest and the smallest equivalence relations on S are:

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