Consider the following statements: 1. A function f:Z→Z, defined by f(x)=x+1, is one-one as well as onto, 2. A function f:N→N, defined by f(x)=x+1, is one-one but not onto. Which of the above statements is\/are correct?
Concept A function f from X to Y is onto (or surjective), if and only if for every element y∈Y there is an element x∈X with f(x)=y. In words: "Each element in the co-domain of has a pre-image" Mathematical Description: f:X→Y is onto⇔∀y‌∃x,f(x)=y One-to-one Correspondence A function f is a one-to-one correspondence (or bijection), if and only if it is both one-to-one and onto In words: "No element in the co-domain of has two (or more) pre images" (one-to-one)and "Each element in the co-domain of f has a pre-image" (onto). Calculation: 1. A function f:Z→Z, defined by f(x)=x+1, is one-one as well as onto. f(x)=x+1 calculate f(x1): f(x1)=x1+1 calculate f(x2): f(x2)=x2+1 Now, f(x1)=f(x2) ⇒x1+1=x2+1 ⇒x1=x2 So, f is one-one function. Consider f(x)=y y=x+1 x=y−1 f(y−1)=y−1+1=y f is onto. 2. A function f:N→N, defined by f(x)=x+1, is one-one but not onto. f(x)=x+1 calculate f(x1) : f(x1)=x1+1 calculate f(x2) : f(x2)=x2+1 Now, f(x1)=f(x2) ⇒x1+1=x2+1 ⇒x1=x2 So, f is one-one function. Clearly, f(x)=x+1≥2 for all x∈N So, f(x) does not assume values 1 . f is not an onto function. So, Both 1 and 2 are correct.