Q. For the set N of natural numbers and a binary operation f : N x N → N, an element z ∊ N is called an identity for f, if f (a, z) = a = f(z, a), for all a ∊ N. Which of the following binary operations have an identity?

1. f (x, y) = x + y - 3
2. f (x, y) = max(x, y)
3. f (x, y) = x^y

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

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Q. Consider the following first order logic formula in which R is a binary relation symbol. ∀x∀y (R(x, y)  => R(y, x)) The formula is

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Q. Let P, Q and R be sets let Δ denote the symmetric difference operator defined as PΔQ = (P U Q) - (P ∩ Q). Using Venn diagrams, determine which of the following is/are TRUE? PΔ (Q ∩ R) = (P Δ Q) ∩ (P Δ R) P ∩ (Q ∩ R) = (P ∩ Q) Δ (P Δ R)

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Q. What is the cardinality of the set of integers X defined below? X = {n | 1 ≤ n ≤ 123, n is not divisible by either 2, 3 or 5} ?

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Q. Let A = {a, b, c, d }, B = { p, q, r, s } denote sets. R : A –> B, R is a function from A to B. Then which of the following relations are not functions ?

(i) { (a, p) (b, q) (c, r) }
(ii) { (a, p) (b, q) (c, s) (d, r) }
(iii) { (a, p) (b, s) (b, r) (c, q) }

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Q. Let A = { 1,2,3,4,…….∞ } and a binary operation ‘+’ is defined by a + b = ab ∀ a,b ∈ A. Which of the following is true ?

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Q. The minimum number of cards to be dealt from an arbitrarily shuffled deck of 52 cards to guarantee that three cards are from some same suit is

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Q. An n x n array v is defined as follows:

v[i, j] = i-j for all i, j, 1 <= i <= n, 1 <= j <= n

The sum of the elements of the array v is

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Q. X, Y and Z are closed intervals of unit length on the real line. The overlap of X and Y is half a unit. The overlap of Y and Z is also half a unit. Let the overlap of X and Z be k units. Which of the following is true?

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Q. E1 and E2 are events in a probability space satisfying the following constraints:

Pr(E1) = Pr(E2)
Pr(EI U E2) = 1
E1 and E2 are independent

The value of Pr(E1), the probability of the event E1 is

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Q. A polynomial p(x) satisfies the following:

p(1) = p(3) = p(5) = 1
p(2) = p(4) = -1

The minimum degree of such a polynomial is

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Q. A relation R is defined on the set of integers as xRy if f(x + y) is even. Which of the following state­ments is true?

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Q. Let a, b, c, d be propositions. Assume that the equivalences a ↔ (b V-b) and b ↔ c hold. Then the truth value of the formula (a ∧ b) → (a ∧ c) ∨ d) is always

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Q. Let G be an undirected connected graph with distinct edge weight. Let emax be the edge with maximum weight and emin the edge with minimum weight. Which of the following statements is false?

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Q. Let G be an undirected graph. Consider a depth-first traversal of G, and let T be the resulting depth-first search tree. Let u be a vertex in G and let v be the first new (unvisited) vertex visited after visiting u in the traversal. Which of the following statements is always true?

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Q. Consider the following statements:

S1: The sum of two singular n × n matrices may be non-singular
S2: The sum of two n × n non-singular matrices may be singular.

Which of the following statements is correct?

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Q. Let f(n) = n^2Logn and g(n) = n (logn)^10 be two positive functions of n. Which of the following statements is correct?

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Q. How many 4-digit even numbers have all 4 digits distinct?

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Q. Let f: A→B be a function, and let E and F be subsets of A. Consider the following statements about images.

S1: f (E ∪ F) = f (E) ∪ f (F)
S1: f (E ∩ F) = f (E) ∩ f (F)

Which of the following is true about S1 and S2?

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