Q. What is the minimum number of ordered pairs of non-negative numbers that should be chosen to ensure that there are two pairs (a, b) and (c, d) in the chosen set such that "a ≡ c mod 3" and "b ≡ d mod 5" .

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Q. Consider the binary relation:

S = {(x, y) | y = x+1 and x, y ∈ {0, 1, 2, ...}}

The reflexive transitive closure of S is

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Q. The following is the incomplete operation table a 4-element group.
 *  e  a  b  c
 e  e  a  b  c
 a  a  b  c  e
 b
 c
The last row of the table is

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Q. The inclusion of which of the following sets into

S = {{1, 2}, {1, 2, 3}, {1, 3, 5}, (1, 2, 4), (1, 2, 3, 4, 5}}

is necessary and sufficient to make S a complete lattice under the partial order defined by set containment ?

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Q. Consider the set ∑* of all strings over the alphabet ∑ = {0, 1}. ∑* with the concatenation operator for strings

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Q. Let (5, ≤) 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 ?

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Q. Let f : A → B be an injective (one-to-one) function.

Define g : 2^A → 2^B as :
g(C) = {f(x) | x ∈ C}, for all subsets C of A.
Define h : 2^B → 2^A as :
h(D) = {x | x ∈ A, f(x) ∈ D}, for all subsets D of B.

Which of the following statements is always true ?

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Q. Which of the following is true?

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Q. The binary relation S = ф (empty set) on set A = {1, 2, 3} is :

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Q. Consider the following relations:

R1(a,b) iff (a+b) is even over the set of integers
R2(a,b) iff (a+b) is odd over the set of integers
R3(a,b) iff a.b > 0 over the set of non-zero rational numbers
R4(a,b) iff |a - b| <= 2 over the set of natural numbers

Which of the following statements is correct?

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Q. Consider the following statements:

S1: There exists infinite sets A, B, C such that
A ∩ (B ∪ C) is finite.
S2: There exists two irrational numbers x and y such
that (x+y) is rational.

Which of the following is true about S1 and S2?

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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 statement is true?

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Q. Let R be the relation on the set of positive integers such that aRb if and only if a and b are distinct and have a common divisor other than 1. Which one of the following statements about R is True?

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Q. The cardinality of the power set of {0, 1, 2 . . ., 10} is _________.

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Q. Consider two relations R1(A, B) with the tuples (1, 5), (3, 7) and R1(A, C) = (1, 7), (4, 9). Assume that R(A,B,C) is the full natural outer join of R1 and R2. Consider the following tuples of the form (A,B,C)

a = (1, 5, null),
b = (1, null, 7),
c = (3, null, 9),
d = (4, 7, null),
e = (1, 5, 7),
f = (3, 7, null),
g = (4, null, 9).

Which one of the following statements is correct?

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Q. The number of onto functions (surjective functions) from set X = {1, 2, 3, 4} to set Y = {a, b, c} is ________________

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Q. Let X and Y denote the sets containing 2 and 20 distinct objects respectively and F denote the set of all possible functions defined from X and Y. Let f be randomly chosen from F. The probability of f being one-to-one is _________.

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Q. Let R be a relation on the set of ordered pairs of positive integers such that ((p, q), (r, s)) ∈ R if and only if p–s = q–r. Which one of the following is true about R?

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Q. Let R1 be a relation from A = {1, 3, 5, 7} to B = {2, 4, 6, 8} and R2 be another relation from B to C = {1, 2, 3, 4} as defined below:

1. An element x in A is related to an element y in B (under R1) if x + y is divisible by 3.
2. An element x in B is related to an element y in C (under R2) if x + y is even but not divisible by 3.

Which is the composite relation R1R2 from A to C?  

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Q. Let f be a function from a set A to a set B, g a function from B to C, and h a function from A to C, such that h(a) = g(f(a)) for all a ∈ A. Which of the following statements is always true for all such functions f and g?  

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