I'm taking Discrete Math and one of my homework problems from Epp's Discrete Mathematics with Applications asks me to prove the following:
If $r$ and $s$ are any two rational numbers, then $\frac{r+s}{2}$ is rational.
It's pretty basic, and here is my proof:
We will use the direct method. Let $r$ and $s$ be rational numbers such that \begin{align}r=\frac{a}{b},\:s=\frac{c}{d},\:\:\:\text{s. th.}\:\:a,b,c,d\in\mathbb{Z}\tag{1}.\end{align} Therefore we have \begin{align}\frac{r+s}{2}=\frac{\left(\frac{a}{b}\right)+\left(\frac{c}{d}\right)}{2}=\frac{ad+cb}{2bd}\tag{2},\end{align}and since the product of two integers is an integer and the sum of two integers is also an integer, $(2)$ is therefore the quotient of two integers, which is by definition a rational number.$\:\:\blacksquare$
Okay, so no problem there. But the next question now asks me to write a corollary to the proof above:
For any rational numbers $r$ and $s$, $2r+3s$ is rational.
I know it is easily proved, but how do I write a corollary to a proof? Do I simply write "corollary:" and then continue? What should be re-stated and what should I assume to be a given from the previous proof? Epp only briefly talks about corollaries on page 168.
Thank you for your time, and please know that I recognize it is not your job to do my homework and nor would I ever ask it of you.
Best Answer
Alright, we have the following theorems given to us from the text.
Theorem 4.2.1: Every integer is a rational number.
Theorem 4.2.2: The sum of any two rational numbers in rational.
Theorem 15 (from exercise 15): The product of any two rational numbers is rational.
Now, question 25 asks derive(prove) a corollary(a word that simply means new theorem that uses the theorems listed above) using previously proved theorems.
Finally, I will establish how such a proof should look and why we call it a corally.
Proof:
If $s$ is rational, then $2s$ is rational. This follows because Theorem 4.2.1 says that every integer is rational so 2 is rational,and Theorem 15 says that the product of any two rational numbers is rational, so $2s$ must be rational.
Furthermore, we know that $3$ is an integer, so by Theorem 4.2.1, 3 is rational. Also, by Theorem 15 we know that $3r$ is rational.
In conclusion, by Theorem 4.2.2, $3r+2s$ is rational.(End of Proof)
Now, if you look at this proof you notice that I have used many other theorems to prove this. Thus we call this specific result a corollary. However, I'm sure I don't have to say this, but just in case, corollaries are just as true as theorems, and we prove them in the same way we prove a theorem.
Let me know if you have any more questions. I hope this helps!