Question

Which of the following functions from A to B are one-one and onto?
(i) f₁ = {(1, 3), (2, 5), (3, 7)} ; A = {1, 2, 3}, B = {3, 5, 7}
(ii) f₂ = {(2, a), (3, b), (4, c)} ; A = {2, 3, 4}, B = {a, b, c}
(iii) f₃ = {(a, x), (b, x), (c, z), (d, z)} ; A = {a, b, c, d,}, B = {x, y, z}

Answer 1

(i) f₁ = {(1, 3), (2, 5), (3, 7)} ; A = {1, 2, 3}, B = {3, 5, 7}

Injectivity:
f₁ (1) = 3
f₁ (2) = 5
f₁ (3) = 7
$\Rightarrow$Every element of A has different images in B.
So, f₁ is one-one.

Surjectivity:
Co-domain of f₁ = {3, 5, 7}
Range of f₁ =set of images = {3, 5, 7}
$\Rightarrow$Co-domain = range
So, f₁ is onto.

(ii) f₂ = {(2, a), (3, b), (4, c)} ; A = {2, 3, 4}, B = {a, b, c}

Injectivity:
f₂ (2) = a
f₂ (3) = b
f₂ (4) = c
$\Rightarrow$Every element of A has different images in B.
So, f₂ is one-one.

Surjectivity:
Co-domain of f₂ = {a, b, c}
Range of f₂ = set of images = {a, b, c}
$\Rightarrow$Co-domain = range
So, f₂ is onto.

(iii) f₃ = {(a, x), (b, x), (c, z), (d, z)} ; A = {a, b, c, d,}, B = {x, y, z}

Injectivity:
f₃ (a) = x
f₃ (b) = x
f₃ (c) = z
f₃ (d) = z
$\Rightarrow$a and b have the same image x. (Also c and d have the same image z)
So, f₃ is not one-one.

Surjectivity:
Co-domain of f₁ ={x, y, z}
Range of f₁ =set of images = {x, z}
So, the co-domain is not same as the range.
So, f₃ is not onto.