Relations and Functions — Class 12 solved problems

39 problems from this chapter, each solved step by step.

Show that an onto function f:1,2,3 →1,2,3 is always one-one.
An onto function f:1,2,3 →1,2,3 is always one-one because for the range to cover all elements of the codomain, each distinct element in the domain must map to a
Let A=-2,-1,0,1,2,3. Define a relation R on A by xRy if and only if y=max(x,1). If l is the number of elements in R, and m and n are the minimum numbers of orde
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32. Show that f: R -> R defined as f(x) = x / sqrt(1 + x²) is one-one but not onto. (function, one-one, onto, real numbers, domain, codomain, range)
The function f(x) = (x)/(√(1 + x²)) is one-one but not onto.
Let A be the set of all students of a boys school. Show that the relation R in A given by R=(a, b): a is sister of b is the empty relation and R^()=(a, b): the
The relation R is an empty relation because no student in a boys' school can be a sister. The relation R^() is a universal relation because the height differenc
Consider the identity function I_N: N → N defined as I_N(x)=x x N. Show that although I_N is onto but I_N+I_N: N → N defined as (I_N+I_N)(x)=I_N(x)+I_N(x)=x+x=2
The identity function I_N(x) = x is onto because its range is N, which is equal to its codomain. However, the function (I_N+I_N)(x) = 2x is not onto because its
Prove that the relation R in the set A = 1, 2, 3, 4, 5 given by R = (a, b): |a - b| is even is an equivalence relation.
The relation R is an equivalence relation.
Let f(x) = log_e x and g(x) = (x^4 - 2x³ + 3x² - 2x + 2)/(2x² - 2x + 1). Then the domain of f ° g is
The domain of f ° g is (-∞, ∞) or R.
Let the domain of the function f(x) = cos^(-1) ( (4x + 5)/(3x - 7)) be [α, β] and the domain of g(x) = log_2(2 - 6log_7(2x + 5)) be [, ]. Then |7(α + β) + 4( +)
| -102 + 2 · 7^(1/3) |
Let A = 1, 2, 3,, 10 and R be a relation on A such that R = (a, b): a = 2b + 1. Let (a_1, a_2), (a_2, a_3), (a_3, a_4),, (a_k, a_k+1) be a sequence of k element
2
Let f: N → Y be a function defined as f(x)=4 x+3, where, Y=y N: y=4 x+3 for some x N. Show that f is invertible. Find the inverse.
The inverse function is f^(-1)(y) = (y-3)/(4).
The relation R = (x, y): x, y Z and x + y is even is:
The relation R is an equivalence relation.
Show that a one-one function f:1,2,3 →1,2,3 must be onto.
A one-one function f:1,2,3 →1,2,3 must be onto because the number of distinct images equals the number of elements in the codomain, making the range equal to th
Let A=1,2,3. The number of relations on A, containing (1,2) and (2,3), which are reflexive and transitive but not symmetric, is:
5
Consider a function f:[0, (π)/(2)] → R given by f(x)=sin x and g:[0, (π)/(2)] → R given by g(x)=cos x. Show that f and g are one-one, but f+g is not one-one.
The functions f(x) = sin x and g(x) = cos x are one-one on [0, (π)/(2)], but their sum (f+g)(x) = sin x + cos x is not one-one on this interval.
Find g of and f ° g, if f: R → R and g: R → R are given by f(x)=cos x and g(x)=3 x². Show that g of ≠ fog.
g ° f(x) = 3cos² x and f ° g(x) = cos(3x²). Since 3cos² x ≠ cos(3x²), it is shown that g ° f ≠ f ° g.
Let A=1,2,3,4 and B=1,4,9,16. Then the number of many-one functions f: A → B such that 1 f(A) is equal to:
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Show that the relation R in the set 1,2,3 given by R=(1,1),(2,2), (3,3),(1,2),(2,3) is reflexive but neither symmetric nor transitive.
The relation R is reflexive but neither symmetric nor transitive.
The function f: (-∞, ∞) → (-∞, 1), defined by f(x) = (2^x - 2^(-x))/(2^x + 2^(-x)), is:
The function is injective but not surjective.
Show that the number of equivalence relation in the set 1,2,3 containing (1,2) and (2,1) is two.
The number of equivalence relations in the set 1,2,3 containing (1,2) and (2,1) is two.
32. Show that f: R → R defined as f(x) = (x)/(√(1+x²)) is one-one but not onto. (function, domain, codomain, one-to-one, onto)
The function f(x) = (x)/(√(1+x²)) is one-one but not onto.
Let X=1,2,3,4,5,6,7,8,9. Let R_1 be a relation in X given by R_1=(x, y): x-y is divisible by 3 and R_2 be another relation on X given by R_2=(x, y):x, y 1,4,7 o
The relations R_1 and R_2 are equal.
Let T be the set of all triangles in a plane with R a relation in T given by R= (T_1, ~T_2): T_1 is congruent to T_2. Show that R is an equivalence relation.
The relation R is an equivalence relation because it is reflexive, symmetric, and transitive.
Let R be the relation defined in the set A=1,2,3,4,5,6,7 by R=(a, b): both a and b are either odd or even. Show that R is an equivalence relation. Further, show
The relation R is an equivalence relation because it is reflexive, symmetric, and transitive. All elements within the subset 1,3,5,7 are related to each other (
Let R be a relation on the set A of ordered pairs of positive integers defined by (x, y) R(u, v) if and only if x v=y u. Show that R is an equivalence relation.
The relation R is an equivalence relation because it is reflexive, symmetric, and transitive.
Let A = -3, -2, -1, 0, 1, 2, 3. Let R be a relation on A defined by xRy if and only if 0 ≤ x² + 2y ≤ 4. Let be the number of elements in R and m be the minimum
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The number of relations on the set A = 1, 2, 3 containing at most 6 elements including (1, 2), which are reflexive and transitive but not symmetric, is _____
5
Let A be the set of all functions f: Z → Z and R be a relation on A such that R = (f,g): f(0)=g(1) and f(1)=g(0). Then R is:
The relation R is symmetric but neither reflexive nor transitive.
Let A=1,2,3. Then show that the number of relations containing (1,2) and (2,3) which are reflexive and transitive but not symmetric is three.
The number of relations containing (1,2) and (2,3) which are reflexive and transitive but not symmetric is three.
Let L be the set of all lines in a plane and R be the relation in L defined as R= (L_1, ~L_2): L_1 is perpendicular to L_2. Show that R is symmetric but neither
The relation R is symmetric but neither reflexive nor transitive.
Let X=R × R. Define a relation R on X as: (a_1, b_1) R (a_2, b_2) b_1=b_2. Statement I: R is an equivalence relation. Statement II: For some (a, b) X, the set S
Statement I is true but Statement II is false.
The number of non-empty equivalence relations on the set 1, 2, 3 is:
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Let R = (1, 2), (2, 3), (3, 3) be a relation defined on the set 1, 2, 3, 4. Then the minimum number of elements, needed to be added in R so that R becomes an eq
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Let f: X → Y be a function. Define a relation R in X given by R=(a, b): f(a)=f(b). Examine whether R is an equivalence relation or not.
The relation R is an equivalence relation.
Let f:2,3,4,5 →3,4,5,9 and g:3,4,5,9 →7,11,15 be functions defined as f(2)=3, f(3)=4, f(4)=f(5)=5 and g(3)=g(4)=7 and g(5)=g(9)=11. Find g ° f.
g ° f = (2,7), (3,7), (4,11), (5,11)
Let A = 0, 1, 2, 3, 4, 5. Let R be a relation on A defined by (x, y) R if and only if maxx, y 3, 4. Then among the statements (S₁): The number of elements in R
Only statement (S 2) is correct.
Is it true that x=e^(log x) for all real x?
No, it is not true that x=e^(log x) for all real x. It is only true for x > 0.
Show that the function f: R → R, defined as f(x)=x², is neither one-one nor onto.
The function f(x) = x² is neither one-one nor onto.
Show that the function f: N → N, given by f(1)=f(2)=1 and f(x)=x-1, for every x>2, is onto but not one-one.
The function f is onto but not one-one.
Show that f: N → N, given by f(x)= & x+1, if x is odd, & x-1, if x is even is both one-one and onto. The image of 1 and -1 under f is 1. Fig 1.4 Solution Suppos
The function f: N → N given by f(x)= x+1, & if x is odd x-1, & if x is even is both one-one and onto.

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