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**scottie.mcdonald** Define g: Integers to Integers by g(n)=4n-5. Is g injective? surjective? justify your answer

injectivity

suppose g(n)=g(m) for some m element of integers

$\displaystyle \Rightarrow $ 4n-5=4m-5

$\displaystyle \Rightarrow $ 4n=4m

$\displaystyle \Rightarrow $ n=m

so therefor this statement is injective since each element of g(n goes to an element of g(m)

for surjectivity i know i have to let b E integers and let a E integers since g:integers to integers and i know i have to show that f(a)=b. so:

f(a)=b

4a-5=b

a=$\displaystyle \frac{b+5}{4} $

$\displaystyle f(a)= 4 \frac{b+5}{4} - 5 $

$\displaystyle = \frac{4b+20}{4} - 5 $

=b+5-5

=b

this is wrong, but i don't understand why. i tried following the books example, but i guess it only works for their specific example.

how am i to show a generic proof for surjectivity in this case? i know that f(a)=2 has no solution in integers (from what my professor wrote on my paper). as well, how do i write the integers symbol using latex?

thank you,

Scott