Question

Let a function `f: N rightarrow N` be defined by

f(n) = `{:[(2n",", n = 2","  4","  6","  8","......),(n - 1",", n = 3","  7","  11","  15","......),((n + 1)/2",", n = 1","  5","  9","  13","......):}`

then f is ______.

विकल्प

• one-one but not onto

• onto but not one-one

• neither one-one nor onto

• one-one and onto

Answer 1

Let a function `f: N rightarrow N` be defined by

f(n) = `{:[(2n",", n = 2","  4","  6","  8","......),(n - 1",", n = 3","  7","  11","  15","......),((n + 1)/2",", n = 1","  5","  9","  13","......):}`

then f is one-one and onto.

Explanation:

Given function is f(x) = `{:[(2n";", n = 2","  4","  6","  8","......),((n - 1)";", n = 3","  7","  11","  15","......),(((n + 1)/2)";", n = 1","  5","  9","  13","......):}`

When n = 2, 4, 6, then 2n is the multiple of 4,

When n = 3, 7, 11, 15 then (n – 1) is not multiple of 4.

When n = 1, 5, 9, 13, then `((n + 1)/2)` is the odd number.

Every number gives exactly one value.

Thus, f is one-one and onto.