Let A = {– 1, 0, 1, 2}, B = {– 4, – 2, 0, 2} and f, g : A → B be functions defined by f(x) = x2 – x, x ∈ A and g(x) = 2|x – ½| – 1, x ∈ A. Are f and g equal? Justify your answer. (Hint: One may note that two functions f : A → B and g : A → B such that f(a) = g (a) ∀ a ∈ A, are called equal functions).

solution:

Given capacities are: f(x) = x2 – x and g(x) = 2|x – ½| – 1

At x = – 1

f(- 1) = 12 + 1 = 2 and g(- 1) = 2|-1 – ½| – 1 = 2

At x = 0

F(0) = 0 and g(0) = 0

At x = 1

F(1) = 0 and g(1) = 0

At x = 2

F(2) = 2 and g(2) = 2

So we can see that, for each a ∈ A , f(a) = g(a) This infers f and g are equivalent capacities.