If N denote the set of all natural numbers and R be the relation

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31.

Which of the following proposition is a tautology?

  • ~ P  ~ q  p  ~ q

  • ~ q  p  ~ q

  • ~ P   p  ~ q

  • ~ P  ~ q  p  ~ q


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32.

If N denote the set of all natural numbers and R be the relation on N × N defined by (a, b) R (c, d), if ad(b + c) = bc(a + d), then R is

  • symmetric only

  • reflexive only

  • transitive only

  • an equivalence relation


D.

an equivalence relation

For (a, b), (c, d)  N × N

(a, b) R (c, d)

 ad(b + c) = ba(a + b)

Reflexive: Since, ab(b + a) = ba(a + b), ∀ ab ∈ N

 (a, b) R (a, b)

So, R is reflexive.

Symmetric: For (a, b), (c, d)  N × N

Let (a, b) R (c, d)

      ad(b + c) = bc(a +d)      bc(a +d) = ad(b + c) cd(d +a) = da(c + b) (c, d) R (a, b)

So, R is symmetric.

Transitive: For(a, b), (c, d), (e, f) ∈ N × N

Let (a, b) R (c, d), (c, d) R (e, f)

ad(b + c) = bc(a + d) , cf(d + e) = de(c + f)

 adb + adc = bca + bcd             ...(i)and cfd + cfe = dec + def              ...(ii)

On multiplying Eq. (i) by ef and Eq. (ii) by ab, then we get

adbef + adcef + cfdab + cfeab

 = bcaef + bcdef + decab + defab

 adcf(b +e)= bcde(a +f)    af(b +e) = be(a +f) (a, b) R (e, f)

So, R is transitive.

Hence R is an equivalence relation.


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33.

The function f : R  R defined by f (x) = (x - 1) (x - 2) (x - 3) is

  • one-one but not onto

  • onto but not one-one

  • both one-one and onto

  • neither one-one nor onto


34.

The relation R defined on the set of natural numbers as { (a, b) : a differs from b by 3} is given

  • {(1, 4), (2, 5), (3, 6), ... }

  • {(4, 1), (5, 2), (6, 3), ... }

  • {(1, 3), (2, 6), (3, 9), ... }

  • None of the above


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35.

If R be a realtion from A = {1, 2, 3, 4} to B = {1, 3, 5} such that (a, b) ∈R  a < b, then ROR-1 is

  • {(1, 3), (1, 5), (2, 3), (2, 5), (3, 5), (4, 5)}

  • {(3, 1), (5, 1), (3, 2), (5, 2), (5, 3), (5, 4)}

  • {(3, 3), (3, 5), (5, 3), (5, 5)}

  • {(3, 3), (3, 4), (4, 5)}


36.

Let f : R R be defined as f(x) = x2 + 1, find f-1(- 5).

  • ϕ

  • ϕ

  • 5

  • - 5, 5


37.

If F is function such that F (0) = 2, F(1) = 3, F(x + 2)= 2F(x) - F(x + 1) for x > 0, then F(5) is equal to

  • - 7

  • - 3

  • 17

  • 13


38.

Let S be a set containing n elements. Then, number of binary operations on S is

  • nn

  • 2n2

  • nn2

  • n2


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39.

If f : [2, 3]  R is defined by f(x) = x3 + 3x - 2, then the range of f(x) is contained in the interval

  • [1, 12]

  • [12, 34]

  • [35, 50]

  • [- 12, 12]


40.

If R be a relation defined as aRb iff a - b > 0, then the relation is

  • reflexive

  • symmetric

  • transitive

  • symmetric and transitive


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