1
vote
2answers
42 views

Do we simplify the Proof by Contradiction?

Prove the following by contradiction: Suppose $a,b\in\mathbb{Z}$. If $4|\left(a^2+b^2\right)$, then $a$ and $b$ are not both odd (in other words, $a$ and $b$ are even) So, I did this: Assume $a$ ...
1
vote
1answer
62 views

Help prove $f:X \rightarrow Y$ is an injection $\Leftrightarrow$ $f:X\rightarrow Y$ is a surjection when $|X|=|Y|$

I need to prove: Given non-empty finite sets $X$ and $Y$ with $|X| = |Y|,$ a function $X\rightarrow Y$ is an injection if and only if it is a surjection. The hint given is to use the pigeonhole ...
0
votes
1answer
232 views

continuous and bounded function without maximum or minimum

Give an example that contradicts this sentence : $f:(0,1]\to\Bbb R$ is a continuous and bounded function in $(0,1]$ then : $f$ has maximum or minimum. I have understood that $\sin(1/x)$ could be a ...
4
votes
1answer
192 views

Nonexistence of a strongly multiplicative increasing function with $f(2)=3$

Show that there does not exist a strictly increasing function $f:\mathbb{N}\rightarrow\mathbb{N}$ satisfying $$f(2)=3$$ $$f(mn)=f(m)f(n)\forall m,n\in\mathbb{N}$$ Progress: Assume the ...