Discrete math functions assignment

**aloufy** · November 13th 2007, 07:58 PM

This is my last assignment for the discrete maths course. I am very swamped with work from other courses. Its due on Friday and don't know how to do the problems. Can someone recommend a good textbook to follow or website that migh help with discrete? The one we are using is DISCRETE MATHS by Goodaire and its not the best.

Any help with the attached problems will be greatly appreciated. Thanks

**kalagota** · November 14th 2007, 06:32 AM

2) a) $(-\infty, 4]$
b) $[1, +\infty)$

3) what was your method??

4)
Let $f:A \rightarrow B$ and $g:B \rightarrow C$ be 1-1.. show that $g \circ f$ is 1-1..

let $x,y \in A$ such that $(g \circ f) (x) = (g \circ f) (y)$ , i.e $g(f(x)) = g(f(y))$

since g is 1-1, then f(x) = f(y).
since f is 1-1, then x=y.
therefore, $g \circ f$ is 1-1. QED

**aloufy** · November 15th 2007, 05:59 AM

Originally Posted by kalagota

2) a) $(-\infty, 4]$
b) $[1, +\infty)$

3) what was your method??

4)
Let $f:A \rightarrow B$ and $g:B \rightarrow C$ be 1-1.. show that $g \circ f$ is 1-1..

let $x,y \in A$ such that $(g \circ f) (x) = (g \circ f) (y)$ , i.e $g(f(x)) = g(f(y))$

since g is 1-1, then f(x) = f(y).
since f is 1-1, then x=y.
therefore, $g \circ f$ is 1-1. QED

Thank you. method for question 3 is by inspection same as question 2. Also could you explain to me how you got to your answers in question 2 and if you do 3 how you go about solving it too? Thanks

**kalagota** · November 15th 2007, 06:42 AM

for number 2 a)

y= 4 - x^2

you know that the range of y=x^2 is [0, +inf)...
so, the range of y=-x^2 is (-inf, 0]..
so if i add 4 to -x^2, the range would be (-inf, 4]

computationally, this is it..

$0 \leq x^2 < \infty$ , so if i multiply -1 to all sides, the inequalities will change, i.e

$-(0) \geq -x^2 > -\infty$ or $-\infty < -x^2 \leq 0$

if i add 4, the neg inf won't be affected since 4 is negligible with a very very "large" number negatively, therefore

$-\infty < 4 - x^2 \leq 4$ ..

just do the same thing with b)

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