Which of the following functions are injective surjective bi

Which of the following functions are injective, surjective, bijective ? (a) f: Z rightarrow Z: x rightarrow x; (c) f: N rightarrow N: x rightarrow + 1 (e) Age (b) f: Z rightarrow Z: x rightarrow x + 1 (d) Social Security Number (f) f: R rightarrow R: x rightarrow x^2

Solution

A function is injective (one-to-one) if every element of the co-domain is mapped to by at most one element of the domain. The function is surjective (onto) if every element of the co-domain is mapped to by at least one element of the domain. The function is bijective (one-to-one and onto) if every element of the co-domain is mapped to by exactly one element of the domain.

(a) f: Z Z : x x is bijective as every element of the co-domain is mapped to by exactly one element of the domain namely itself.

(b) The function is bijective as every element of the co-domain Z is mapped to by exactly one element of the domain. Z.

(c) Some mathematicians opine that the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3,… If N includes 0, then 1 has the pre-image 0, otherwise 1 does not have any pre-image. Even if 0 is included in N, the number 0 does not have any pre-image in N as -1 does not belong to N. The function f is, therefore, not surjective. However, the function is injective as no element of the co-domain has more than one pre-image. Also, every element of the domain has exactly one image in the co-domain. Apparently, then f is not bijective as it is not surjective

(d) The function f is bijective as the social security number is a unique 9 digit number issued to every US citizen. There is exactly one social security number for an individual. the vice-versa is also true, i.e. every US citizen has a unique social security number. Therefore, f is injective, surjective and bijective.

(e) Every person has a unique age but there may be several people of the same age. Therefore, the function is surjective but not injective and therefore, not bijective.

(e) Every positive element x of R has two pre-images namely, ± x. Thus the function f is not injective. Since the negative real numbers do not have a square root in R, the function is also not surjective and hence not bijective also.