The two cases we must prove are x is even and x is odd. We will prove \(ax^2+bx+c\) is odd for both cases to show that it cannot be equal to zero.
We will use the previously proved proposition "Two integers have the same parity if and only if their sum is even" to show that the sum of an even and an odd number is odd.
If the previous propositon is true, then the proposition "two integers have opposite parity if and only if their sum is odd". (\(P \leftrightarrow Q \equiv \neg P \leftrightarrow \neg Q\))
We will then use this statement and its negation to show that the sum of three odds is odd. this is equivilant to two odds added, which is even, added to another odd which is the opposite parity making the total sum odd.
(i) If x is even, then \(x=2k\).
\[
\begin{align}
ax^2+bx+c &= a(2k)^2+b \cdot 2k+c \\
&=4k^2a+2kb+c \\
&=2 \cdot(k^2a+kb)+c\\
\end{align}
\]
Since this takes the form \(2 \cdot int+odd\), \(ax^2+bx+c \) is odd
(ii) If x is odd, \(x=2k+1\).
\[
\begin{align}
ax^2+bx+c &= a(2k+1)^2+b(2k+1)+c\\
&=a(4k^2+4k+1)+2kb+b+c\\
&=(a+b+c)+2 \cdot(2k^2a+2ka+kb)
\end{align}
\]
Therefore \(2 \cdot(2k^2a+2ka+kb)+(a+b+c)\) is in the form \(2 \cdot int+odd\) and is odd
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For every n ∊ N, (1-(1/2))(1-(1/4))(1-(1/8))(1-(1/16))...(1-(1/2)) is greater than or equal to (1/4) +(1/(2^(n+1)))
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This looks great! Here are some suggestions to perfect it. This might look like a lot, but I am being a stickler so that you can make it absolutely perfect. Your proof is already in quite nice shape.
ReplyDeleteSubstantive suggestions:
(1) Make sure all sentences are truly complete sentences. For example, "Prove..." should become "We will prove...".
(2) "Prove ax^2+bx+c is odd for both cases since zero isn't an odd number." - This would benefit from rephrasing, since as it is, it sounds as though ax^2+bx+c will be odd because of the fact that 0 is not odd.
(3) To justify the fact that the sum of an even and an odd integer is odd, refer to previous result https://proofwiki.blogspot.com/2021/09/if-two-integers-have-same-parity-their.html, and similarly to show that the sum of three odd numbers is odd.
(4) Where does the statement 3(2k+1)=6k+3) get used in the proof? The next to last sentence could be clarified.
Formatting suggestions:
(1) Since \(*\) has other definitions than multiplication of two reals in lots of fields of math, I recommend using \(\cdot \) (\cdot) instead
(2) The last line of displayed math has an extra plus sign.
I tried to fix what you suggested let me know if there are any other changes I should make.
DeleteThese are excellent improvements. I now have only small typo/spellcheck sorts of suggestions. I notice that a couple of times 'proposition' appears as 'propisition' and 'opposite' appears as 'oppisite'. 'Statement' appears as 'statment' and 'sum' appears as 'summ'. Additionally, 'it's negation' should be 'its negation' (no apostrophe). In 'the sum of and even and an odd...' the first 'and' should be 'an'. I also note an extra + sign in the final line of displayed math.
DeleteAs a small note, usually when you carry on a calculation over several steps, rather than repeating the previous form on the left hand side of the equality on the next line, you align the equals signs and keep going. E.g. \[ \begin{align} ax^2 + bx + c &= a(2k)^2 + b(2k) + c \\ &= 2 (2 a k^2 + bk) + c \end{align}
Very nice proof.