S.T. function f : N → N by f (1) = f(2) = 1 and f(x) = x - 1 for eve

1.	S.T. function f : N → N by f (1) = f(2) = 1 and f(x) = x - 1 for every x > 2, is on to but not one-one.
Answer» f: N → N by f(1) = f(2) = 1 and f(x) = x - 1. f is not one-one because f (1) = 1 and f(2) = 1 ∴ f(1) = f(2) but 1 ≠ 2 ∴ f is not one-one for every y ∈ N then f(x) = y - x - 1 then y = x - 1 ⇒ x ∈ N ∴ y ∈ N ∋ x ∈ N ∴ f is onto.

S.T. function f : N → N by f (1) = f(2) = 1 and f(x) = x - 1 for every x > 2, is on to but not one-one.

Answer»

f: N → N by f(1) = f(2) = 1 and f(x) = x - 1.

f is not one-one because

f (1) = 1 and f(2) = 1

∴ f(1) = f(2)

but 1 ≠ 2

∴ f is not one-one

for every y ∈ N then f(x) = y - x - 1 then y = x - 1

⇒ x ∈ N

∴ y ∈ N ∋ x ∈ N

∴ f is onto.

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S.T. function f : N → N by f (1) = f(2) = 1 and f(x) = x - 1 for every x &gt; 2, is on to but not one-one.

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S.T. function f : N → N by f (1) = f(2) = 1 and f(x) = x - 1 for every x > 2, is on to but not one-one.