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181.
Let X = {1, 2, 3, 4, 5}. The number of different ordered pairs (Y, Z) that can be formed such that Y ⊆ X, Z ⊆ X and Y ∩ Z is empty, is

5²

3⁵

2⁵

2⁵

3⁵

Y ⊆ X, Z ⊆ X
Let a ∈ X, then we have following chances that
(1) a ∈ Y, a ∈ Z
(2) a ∉ Y, a ∈ Z
(3) a ∈ Y, a ∉ Z
(4) a ∉ Y, a ∉ Z
We require Y ∩ Z = φ
Hence (2), (3), (4) are chances for ‘a’ to satisfy Y ∩ Z = φ.
∴ Y ∩ Z = φ has 3 chances for a.
Hence for five elements of X, the number of required chance is
3 × 3 × 3 × 3 × 3 = 3⁵

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182.

Let R be the set of real numbers.
Statement-1 : A = {(x, y) ∈R × R : y - x is an integer} is an equivalence relation on R.
Statement-2 : B = {(x, y) ∈ R × R : x = αy for some rational number α} is an equivalence relation on R.

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(2) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
Statement-1 is true, Statement-2 is false.
Statement-1 is true, Statement-2 is false.

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183.

If A, B and C are three sets such that A ∩ B = A∩ C and A ∪ B = A ∪ C, then

A = B
A = C
B = C
B = C

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184.

A die is thrown. Let A be the event that the number obtained is greater than 3. Let B be the event that the number obtained is less than 5. Then P (A ∪ B) is

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185.

Let A and B be two events such that $straight P space left parenthesis top enclose straight A union straight B end enclose right parenthesis space equals space 1 over 6 comma space straight P space left parenthesis straight A intersection straight B right parenthesis space equals space 1 fourth space and space straight P space left parenthesis top enclose straight A right parenthesis space equals space 1 fourth comma$ where $top enclose straight A$ stands for complement of event A. Then events A and B are

equally likely and mutually exclusive
equally likely but not independent
independent but not equally likely
independent but not equally likely

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186.

Two sets A and B are as under:

A = {(a-b)∈ RxR:|a-5|<1 and |b-5|<1}

B = {(a,b)∈ Rx R: 4(a-6)² + 9 (b-5)² ≤ 36},then

Neither A ⊂ B nor B ⊂ A
B ⊂ A
A ⊂ B
A ∩ B = ∅

187.

On R, the set of real numbers, a relation p is defined as 'a $ρ$ b if and only if 1+ ab> 0'. Tnen,

$ρ$ is an equivalence relation
$ρ$ is reflexive and transitive but not symmetric
$ρ$ is reflexive and symmetric but not transitive
$ρ$ is only symmetric

188.

Let R be a relation defined on the set Z of all integers and xRy, when x + 2y is divisible by 3, then

A is not transitive
R is symmetric only
R is an equivalence relation
R is not an equivalence relation

189.

Let f : N $\to$ R be such that f(1) = 1 and f(1) + 2f(2) + 3f(3) + ... + nf(n) = n(n+ 1) f(n), for all n $\in$ N, n $\geq$ 2, where N is the set of natural numbers and R is the set of real numbers. Then, the value of f(500) is

1000
500
1/500
1/1000

190.

For the function f(x) = $[\frac{1}{[x]}]$ . where [x] denotes the greatest integer less than or equal to x, which of the following statements are true?