Lecture 4 : Revision

Let A = {9, 10, 11, 12, 13} and let f : A → N be defined by f(n) = the highest prime factor of n. Find the range off.

Let f be the subset of Z x Z defined by f = {(ab, a+ b): a, b ∈ Z}. Is f a function from Z to Z: justify your answer.

Let A = {1, 2, 3, 4}, B = {1, 5, 9, 11, 15, 16} and f = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}. Are the following true?

(i) f is a relation from A to B.

(ii) f is a function from A to B.

Justify your answer in each case.

Let R be a relation from N to N defined by R = {(a, b): a, b ∈ N and a = b²}. Are the following true?

(i) (a, a) ∈ R, for all a ∈ N
(ii) (a, b) ∈ R, implies (b, a) ∈ R

(iii) (a, b) ∈ R, (b, c) ∈ R implies (a, c) ∈ R.

Justify your answer in each case.

Let f = {(1, 1), (2, 3), (0, -1), (-1, -3)} be a function from Z to Z defined by f(x) = ax + b, for some integers a, b. Determine a, b.

Letbe a function from R into R. Determine the range of f.

Find the domain and the range of the real function f defined by f(x) = |x- 1|.

The relation f is defined by 

The relation g is defined by 

Show that f is a function and g is not a function.

Find the domain of the function 

If f(x) =  x², find