Let \(A = \{a, b, c\}\) and the relation \(R\) be defined on \(A\) as follows: \(R = \{(a,a),(b,c),(a,b)\}\). Then, write minimum number of ordered pairs to be added in \(R\) to make \(R\) reflexive and transitive.
(b,b), (c,c), (a,c)
Let \(D\) be the domain of the real valued function \(f\) defined by \(f(x) = \sqrt{25 - x^2}\). Then write \(D\).
[-5, 5]
Let \(f, g : \mathbb{R} \to \mathbb{R}\) be defined by \(f(x) = 2x + 1\) and \(g(x) = x^2 - 2\). Find \(g \circ f\).
4x² + 4x − 1
Let \(f : \mathbb{R} \to \mathbb{R}\) be defined by \(f(x) = 2x - 3\). Write \(f^{-1}\).
\(f^{-1}(x) = \dfrac{x + 3}{2}\)
If \(A = \{a,b,c,d\}\) and the function \(f = \{(a,b),(b,d),(c,a),(d,c)\}\), write \(f^{-1}\).
\{(b,a),(d,b),(a,c),(c,d)\}
If \(f : \mathbb{R} \to \mathbb{R}\) is defined by \(f(x) = x^2 - 3x + 2\), write \(f(f(x))\).
\(x^4 - 6x^3 + 10x^2 - 3x\)
Is \(g = \{(1,1),(2,3),(3,5),(4,7)\}\) a function? If \(g(x) = \alpha x + \beta\), then what are the values of \(\alpha\) and \(\beta\)?
\(\alpha = 2, \beta = -1\)
Are the following sets of ordered pairs functions? If so, examine whether the mapping is injective or surjective.
(i) \(\{(x,y): x \text{ is a person}, y \text{ is the mother of } x\}\)
(ii) \(\{(a,b): a \text{ is a person}, b \text{ is an ancestor of } a\}\)
(i) Represents a function which is surjective but not injective.
(ii) Does not represent a function.
If mappings \(f\) and \(g\) are given by \(f = \{(1,2),(3,5),(4,1)\}\) and \(g = \{(2,3),(5,1),(1,3)\}\), write \(g \circ f\).
\{(2,5),(5,2),(1,5)\}
Let \(C\) be the set of complex numbers. Prove that the mapping \(f:C \to \mathbb{R}\) defined by \(f(z)=|z|\) is neither one-one nor onto.
Neither one-one nor onto.
Let the function \(f:\mathbb{R} \to \mathbb{R}\) be defined by \(f(x) = \cos x\). Show that \(f\) is neither one-one nor onto.
Neither one-one nor onto.
Let \(X = \{1,2,3\}\) and \(Y = \{4,5\}\). Determine whether the following sets of \(X \times Y\) are functions from \(X\) to \(Y\):
(i) \(f = \{(1,4),(1,5),(2,4),(3,5)\}\)
(ii) \(g = \{(1,4),(2,4),(3,4)\}\)
(iii) \(h = \{(1,4),(2,5),(3,5)\}\)
(iv) \(k = \{(1,4),(2,5)\}\)
(i) f is not a function
(ii) g is a function
(iii) h is a function
(iv) k is not a function
If functions \(f:A \to B\) and \(g:B \to A\) satisfy \(g \circ f = I_A\), show that \(f\) is one-one and \(g\) is onto.
f is one-one and g is onto.
Let \(f:\mathbb{R} \to \mathbb{R}\) be defined by \(f(x) = \dfrac{1}{2 - \cos x}\). Find the range of \(f\).
\([1/3, 1]\)
Let \(n\) be a fixed positive integer. Define a relation \(R\) in \(\mathbb{Z}\) as follows: for \(a,b \in \mathbb{Z}\), \(aRb\) iff \(a-b\) is divisible by \(n\). Show that \(R\) is an equivalence relation.
R is an equivalence relation.
Given \(A = \{1,2,3,4\}\), define relations on \(A\) having the properties:
Let \(R\) be a relation defined on the set of natural numbers \(\mathbb{N}\) as follows:
\[ R = \{(x,y) : x,y \in \mathbb{N},\ 2x + y = 41 \}. \]
Find the domain and range of the relation \(R\). Also verify whether \(R\) is reflexive, symmetric and transitive.
Domain of \(R = \{1,2,3,4,\ldots,20\}\)
Range of \(R = \{1,3,5,7,9,\ldots,39\}\)
\(R\) is neither reflexive, nor symmetric, nor transitive.
Given \(A = \{2,3,4\}\) and \(B = \{2,5,6,7\}\), construct an example of each of the following:
Give an example of a map:
Let \(A = \mathbb{R} - \{3\}\) and \(B = \mathbb{R} - \{1\}\). Let \(f : A \to B\) be defined by
\[ f(x) = \dfrac{x - 2}{x - 3}. \]
Show that \(f\) is bijective.
Let \(A = [-1,1]\). Discuss whether the following functions defined on \(A\) are one-one, onto or bijective:
(i) \(f(x) = \dfrac{x}{2}\)
(ii) \(g(x)=|x|\)
(iii) \(h(x) = x|x|\)
(iv) \(k(x)=x^2\)
(i) \(f\) is one-one but not onto.
(ii) \(g\) is neither one-one nor onto.
(iii) \(h\) is bijective.
(iv) \(k\) is neither one-one nor onto.
Each of the following defines a relation on \(\mathbb{N}\):
(i) \(x\) is greater than \(y\)
(ii) \(x + y = 10\)
Determine which of the above relations are reflexive, symmetric and transitive.
(i) Transitive
(ii) Symmetric
(iii) Reflexive, symmetric and transitive
(iv) Transitive
Let \(A = \{1,2,3,\ldots,9\}\) and \(R\) be the relation defined on \(A \times A\) by
\[(a,b) R (c,d) \text{ if } a+d = b+c.\]
Prove that \(R\) is an equivalence relation and find the equivalence class \([(2,5)]\).
\([(2,5)] = \{(1,4),(2,5),(3,6),(4,7),(5,8),(6,9)\}
Using the definition, prove that the function \(f : A \to B\) is invertible if and only if \(f\) is both one-one and onto.
Functions \(f,g : \mathbb{R} \to \mathbb{R}\) are defined by \(f(x)=x^2+3x+1\) and \(g(x)=2x−3\). Find:
(i) \(f \circ g\)
(ii) \(g \circ f\)
(iii) \(f \circ f\)
(iv) \(g \circ g\)
(i) \((f \circ g)(x) = 4x^2 - 6x + 1\)
(ii) \((g \circ f)(x) = 2x^2 + 6x - 1\)
(iii) \((f \circ f)(x)= x^4 + 6x^3 + 14x^2 + 15x + 5\)
(iv) \((g \circ g)(x)=4x - 9\)
Let * be the binary operation defined on \(\mathbb{Q}\). Determine which of the following operations are commutative:
(i) \(a * b = a - b\)
(ii) \(a * b = a^2 + b^2\)
(iii) \(a * b = a + ab\)
(iv) \(a * b = (a - b)^2\)
(ii) & (iv)
Let * be a binary operation defined on \(\mathbb{R}\) by \(a * b = 1 + ab\). Then determine whether the operation is:
(i) commutative but not associative
(ii) associative but not commutative
(iii) neither commutative nor associative
(iv) both commutative and associative
(i)
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
Let the relation \(R\) be defined in \(\mathbb{N}\) by \(aRb\) if \(2a + 3b = 30\). Then \(R = \) ____.
\(\{(3,8),(6,6),(9,4),(12,2)\}\)
Let the relation \(R\) be defined on the set \(A=\{1,2,3,4,5\}\) by \(R=\{(a,b):|a^2-b^2|<8\}\). Then \(R\) is given by ____.
\(\{(1,1),(1,2),(2,1),(2,2),(2,3),(3,2),(3,3),(3,4),(4,3),(4,4),(5,5)\}\)
Let \(f=\{(1,2),(3,5),(4,1)\}\) and \(g=\{(2,3),(5,1),(1,3)\}\). Then \(g\circ f =\) ____ and \(f\circ g=\) ____.
\(g\circ f=\{(1,3),(3,1),(4,3)\}\) and \(f\circ g=\{(2,5),(5,2),(1,5)\}\)
Let \(f:\mathbb{R}\to\mathbb{R}\) be defined by \(f(x)=\dfrac{x}{\sqrt{1+x^2}}\). Then \((f\circ f\circ f)(x)=\) ____.
\(\dfrac{x}{\sqrt{3x^2+1}}\)
If \(f(x)=4-(x-7)^3\), then \(f^{-1}(x)=\) ____.
\(7 + (4 - x)^{1/3}\)
Let R = {(3,1), (1,3), (3,3)} be a relation defined on the set A = {1,2,3}. Then R is symmetric, transitive but not reflexive.
False
Let f : \(\mathbb{R} \to \mathbb{R}\) be the function defined by f(x) = sin(3x + 2) for all x ∈ \(\mathbb{R}\). Then f is invertible.
False
Every relation which is symmetric and transitive is also reflexive.
False
An integer m is said to be related to another integer n if m is an integral multiple of n. This relation in \(\mathbb{Z}\) is reflexive, symmetric and transitive.
False
Let A = {0, 1} and N be the set of natural numbers. Then the mapping f : N → A defined by f(2n − 1) = 0 and f(2n) = 1 for all n ∈ N is onto.
True
The relation R on the set A = {1, 2, 3} defined as R = {(1,1), (1,2), (2,1), (3,3)} is reflexive, symmetric and transitive.
False
The composition of functions is commutative.
False
The composition of functions is associative.
True
Every function is invertible.
False
A binary operation on a set has always the identity element.
False