Some examples on proving/disproving a function is injective/surjective (CSCI 2824, Spring 2015)This page contains some examples that should help you finish Assignment 6. (See also Section 4.3 of the textbook) Proving a function is injectiveRecall that a function is injective/one-to-one if .
Example 1: Disproving a function is injective (i.e., showing that a function is not injective)Consider the function . (This function defines the Euclidean norm of points in .) Recall also that . Claim: is not injective. Proof
Note that are distinct and . Hence is not injective. QED. Example 2: Proving a function is injectiveConsider the function . Claim: is injective. Proof
Proving a function is surjectiveRecall that a function is surjectiveonto if .
Example 3: disproving a function is surjective (i.e., showing that a function is not surjective)Consider the function . Claim: is not surjective. Proof
Example 4: disproving a function is surjective (i.e., showing that a function is not surjective)Consider the absolute value function . Claim: is not surjective. Proof
Example 5: proving a function is surjectiveConsider again the function . Claim: is injective. Scrap work
Proof
Finding the inverseOnce we show that a function is injective and surjective, it is easy to figure out the inverse of that function. The inverse is simply given by the relation you discovered between the output and the input when proving surjectiveness. Only bijective functions have inverses! A very rough guide for finding inverse
If we are given a bijective function , to figure out the inverse of we start by looking at the equation . Then we perform some manipulation to express in terms of . Example 6Consider the function . We claim (without proof) that this function is bijective. So what is the inverse of ?
. Example 7The function that we consider in Examples 2 and 5 is bijective (injective and surjective). The inverse is given by . Note that this expression is what we found and used when showing is surjective. |