Q5. A = {1, 2, 3}, B = {4, 5, 6, 7} and f = {(1, 4), (2, 5), (3, 6)}. Show that f : A → B is one-one. Is it onto? (NCERT Exercise 1.2, Q6)
The images 4, 5 and 6 are all different, so f is one-one. 7 is not an image, so f is not onto: it is an into function.
Q6. For sets A and B, show that f : A × B → B × A, f(a, b) = (b, a), is bijective (NCERT Exercise 1.2, Q8).
If (b, a) = (b, a), then a = a and b = b, so (a, b) = (a, b): one-one. Any (b, a) ∈ B × A is the image of (a, b) ∈ A × B: onto.
Q7. Show that f : R → R, f(x) = x3, is injective (NCERT Miscellaneous Exercise, Q2).
Factorise:
x3 − x3 = (x − x)(x2 + xx + x2).
The second factor equals (x + x/2)2 + 3x2/4, which is 0 only when x = x = 0. So x3 = x3 forces x = x.
Q8. (a) The identity function IN : N → N is onto. Is IN + IN, x ↦ 2x, onto? (b) f(x) = sin x and g(x) = cos x are both one-one on [0, π/2]. Is f + g one-one there? (NCERT Miscellaneous Examples 25 and 26, pages 14–15)
(a) No: 2x = 3 has no solution in N. (b) No: (f + g)(0) = 0 + 1 = 1 and (f + g)(π/2) = 1 + 0 = 1. A sum of onto functions need not be onto, and a sum of one-one functions need not be one-one.
Q9. How many one-one functions are there from {1, 2, 3} to itself, and how many onto functions from {1, 2, 3, …, n} to itself? (NCERT Miscellaneous Example 22 and Miscellaneous Exercise, Q4)
A one-one function from {1, 2, 3} to itself lists the three numbers in some order: 3! = 6. On a finite set onto is the same as one-one, so the onto functions from {1, 2, …, n} to itself are the n! orderings: n!.
Q10. A = {−1, 0, 1, 2}, B = {−4, −2, 0, 2}, f(x) = x2 − x and g(x) = 2|x − 1/2| − 1 for x ∈ A. Are f and g equal? (Two functions from A to B are equal when f(a) = g(a) for every a ∈ A.) (NCERT Miscellaneous Exercise, Q5)
f(−1) = 2, f(0) = 0, f(1) = 0, f(2) = 2. g(−1) = 2 × 3/2 − 1 = 2, g(0) = 2 × 1/2 − 1 = 0, g(1) = 0 and g(2) = 2 × 3/2 − 1 = 2. The values agree at every point of A, so f = g, although the formulas differ.
Q11. A has 3 elements and B has 2. How many functions are there from A to B, and how many of them are one-one or onto? Answer the same for functions from B to A.
A to B: 23 = 8 functions; none is one-one (3 elements cannot have 3 different images in a 2-element set); all but the 2 constant functions are onto, so 6. B to A: 32 = 9 functions; 3 × 2 = 6 are one-one; none is onto, since 2 elements have at most 2 images.
Q12. Is f : R → R, f(x) = x|x|, bijective?
Yes. f(x) = x2 for x ≥ 0 and −x2 for x < 0, and f(x) has the sign of x. So equal images need inputs of the same sign: two inputs ≥ 0 with x2 = x2, or two negative inputs with −x2 = −x2, must be equal. So f is one-one. Any y ≥ 0 is f(√y), and any y < 0 is f(−√(−y)): onto.