Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as aRb if a is congruent to b.
Then R is
reflexive but not transitive
transitive but not symmetric
equivalence
none of these
Consider the non-empty set consisting of children in a family and a relation R defined as aRb if a is a brother of b.
Then R is
symmetric but not transitive
transitive but not symmetric
neither symmetric nor transitive
both symmetric and transitive
The maximum number of equivalence relations on the set A = {1, 2, 3} is
1
2
3
5
If a relation R on the set {1, 2, 3} is defined by R = {(1, 2)}, then R is
reflexive
transitive
symmetric
none of these
Let us define a relation R in ℝ as aRb if a ≥ b.
Then R is
an equivalence relation
reflexive, transitive but not symmetric
symmetric, transitive but not reflexive
neither transitive nor reflexive but symmetric
Let A = {1, 2, 3} and consider the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}.
Then R is
reflexive but not symmetric
reflexive but not transitive
symmetric and transitive
neither symmetric nor transitive
The identity element for the binary operation * defined on ℚ − {0} as a * b = (ab)/2 is
1
0
2
none of these
If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is
720
120
0
none of these
Let A = {1, 2, 3, …, n} and B = {a, b}. Then the number of surjections from A into B is
nP2
2ⁿ − 2
2ⁿ − 1
none of these
Let f : ℝ → ℝ be defined by f(x) = 1/x. Then f is
one-one
onto
bijective
f is not defined
Let f : ℝ → ℝ be defined by f(x) = 3x² − 5 and g : ℝ → ℝ be defined by g(x) = x/(x² + 1).
Then g ∘ f is
(3x² − 5)/(9x⁴ − 30x² + 26)
(3x² − 5)/(9x⁴ − 6x² + 26)
3x²/(x⁴ + 2x² − 4)
3x²/(9x⁴ + 30x² − 2)
Which of the following functions from ℤ into ℤ are bijections?
f(x) = x³
f(x) = x + 2
f(x) = 2x + 1
f(x) = x² + 1
Let f : ℝ → ℝ be defined by f(x) = x³ + 5. Then f⁻¹(x) is
(x + 5)^{1/3}
(x − 5)^{1/3}
(5 − x)^{1/3}
5 − x
Let f : A → B and g : B → C be bijective functions. Then (g ∘ f)⁻¹ is
f⁻¹ ∘ g⁻¹
f ∘ g
g⁻¹ ∘ f⁻¹
g ∘ f
Let f : ℝ − {3/5} → ℝ be defined by f(x) = (3x + 2)/(5x − 3).
Then
f⁻¹(x) = f(x)
f⁻¹(x) = −f(x)
(f ∘ f)(x) = −x
f⁻¹(x) = (1/19)f(x)
Let f : [0,1] → [0,1] be defined by
f(x) = x, if x is rational
f(x) = 1 − x, if x is irrational
Then (f ∘ f)(x) is
constant
1 + x
x
none of these
Let f : [2, ∞) → ℝ be defined by f(x) = x² − 4x + 5. Then the range of f is
ℝ
[1, ∞)
[4, ∞)
[5, ∞)
Let f : ℕ → ℝ be defined by f(x) = (2x − 1)/2 and g : ℚ → ℝ be defined by g(x) = x + 2.
Then (g ∘ f)(3/2) is
1
1
7/2
none of these
Let f : ℝ → ℝ be defined by
f(x) = 2x, x > 3
f(x) = x², 1 ≤ x ≤ 3
f(x) = 3x, x ≤ 1
Then f(−1) + f(2) + f(4) is
9
14
5
none of these
Let f : ℝ → ℝ be given by f(x) = tan x. Then f⁻¹(1) is
π/4
{nπ + π/4 : n ∈ ℤ}
does not exist
none of these