Functions one-one into and onto - One Line Questions

1. Let A = {1, 2} and B = {3, 4}. How many onto functions can be defined from A to B? — 4
2. What is the minimum number of elements required in the codomain B for a function f: A → B to be onto, if |A| = 5? — 5
3. Let A = {1, 2} and B = {3, 4}. How many one-one functions can be defined from A to B? — 4
4. If f: A → B is a function, and A = {1, 2}, B = {3}. How many functions are possible? — 3
5. A function f: A → B is called onto (surjective) if: — The range of f is equal to the codomain B.
6. For a function f: A → B to be one-one, the inverse image of each element in the codomain must be: — Either empty or a single element
7. A function f: A → B is called a bijection if it is: — Both one-one and onto
8. What is the condition for a function f: A → B to be one-one (injective)? — Distinct elements of A are mapped to distinct elements of B.
9. Which of the following describes a function that is neither one-one nor onto? — f(x) = x², f: R → R
10. If f: A → B is one-one, then for any two distinct elements x₁, x₂ ∈ A, we have: — f(x₁) ≠ f(x₂)
11. If f: A → B is an onto function, then which of the following is true? — f⁻¹(y) is defined for all y ∈ B
12. If f: A → B is one-one, then the number of elements in A must be: — Less than or equal to the number of elements in B
13. If f: A → B is one-one and |A| = |B|, then f must be: — Bijective
14. A function f: A → B is called a(n) ______ function if every element of B has exactly one pre-image in A. — Bijective
15. If f: A → B is both one-one and onto, then it is called a(n): — Bijective function
16. Which type of function guarantees that every element in the codomain has at least one pre-image in the domain? — Onto function
17. For a function f: A → B, if |A| = m and |B| = n, and f is one-one, what can be said about m and n? — m ≤ n
18. What is the total number of functions from a set A with m elements to a set B with n elements? — nᵐ
19. If f: A → B is a function such that |A| = n and |B| = m. If f is one-one, then: — n ≤ m
20. If f: A → B is a function such that |A| = n and |B| = m. If f is onto, then: — n ≥ m
21. What is the total number of one-one functions from a set A with m elements to a set B with n elements, where m ≤ n? — P(n, m)
22. What is the total number of onto functions from a set A with m elements to a set B with n elements, where m ≥ n? — n! S(m, n)
23. For a function f: A → B, if f is onto, what can be said about the number of elements in the range and codomain? — Number of elements in Range(f) = Number of elements in B
24. If f: A → B is a function and there exists an element b ∈ B such that f(x) ≠ b for all x ∈ A, then f is: — Into
25. Which of the following statements is true for an 'into' function f: A → B? — Range(f) ⊂ B
26. Which condition must be met for a function f: A → B to be classified as 'into'? — Range(f) ⊂ B and Range(f) ≠ B
27. If a function f: A → B is not onto, it means: — There exists at least one element in B that is not the image of any element in A.
28. If f: A → B is a function, and A = {1, 2}, B = {3}. Is this function one-one? — No, because two elements must map to the same element.
29. If f: R → R is defined by f(x) = 1/x, is this function one-one? — Yes, because 1/x is unique for each x.
30. Let f(x) = 2ˣ, where f: R → R. Is this function onto? — No, because the range is (0, ∞), not R.
31. If A = {1, 2, 3} and B = {a, b, c, d}, and f: A → B is defined as f(1)=a, f(2)=b, f(3)=c. Is f an onto function? — No, because 'd' in B is not mapped.
32. Consider the function f(x) = x mod 3, where f: Z → {0, 1, 2}. Is this function onto? — Yes, because every element in the codomain is obtained.
33. Let f(x) = eˣ, where f: R → R. Is this function onto? — No, because the range is (0, ∞), not R.
34. Let f(x) = eˣ, where f: R → R. Is this function one-one? — Yes, because it is strictly increasing.
35. Consider the function f(x) = 2x + 3, where f: R → R. Is this function one-one? — Yes, because it is a linear function with a non-zero slope.
36. If f: R → R is defined by f(x) = 1/x, is this function onto? — No, because 0 is in the codomain but not in the range.
37. Consider the function f(x) = x+1 for f: Z → Z. Is this function one-one? — Yes, because for every y in Z, there is a unique x in Z such that f(x) = y.
38. Consider the function f(x) = x+1 for f: Z → Z. Is this function onto? — Yes, because for every y in Z, we can find x = y-1 in Z.
39. Let f(x) = sin(x), where f: R → R. Is this function onto? — No, because the range is [-1, 1], not R.
40. If f: A → B is a function, and A = {1, 2}, B = {3}. Is this function onto? — Yes, because the only element in B is mapped to.
41. Let f(x) = x² for f: {-1, 1} → {1}. Is this function onto? — Yes, because the range {1} is equal to the codomain {1}.
42. Let f(x) = x² for f: {-1, 1} → {1}. Is this function one-one? — No, because f(-1) = f(1) = 1.
43. Consider the function f(x) = x², where f: R → R. Is this function onto? — No, because negative numbers are not in the range.
44. Let f(x) = |x|, where f: R → R. Is this function onto? — No, because negative numbers are not in the range.
45. Consider the function f(x) = x mod 3, where f: Z → {0, 1, 2}. Is this function one-one? — No, because f(0) = f(3) = 0.
46. Consider the function f(x) = x², where f: R → R. Is this function one-one? — No, because f(x) = f(-x) for non-zero x.
47. Consider f(x) = x³ where f: R → R. Is this function onto? — Yes, because x³ can take any real value.
48. If f(x) = c (a constant), where f: R → R, is this function one-one? — No, unless the domain has only one element.
49. If f(x) = x for all x in A, where f: A → A, this is called: — Identity function
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