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Oxford MAT 2013/2014

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Reply 220
Avatar for JosephML
JosephML
2
Original post by MEPS1996
x+ sqrt(1-x) <= 5/4
sqrt(1-x)<= 5/4 - x
since we are taking the positive square root, both values are not negative,
so |sqrt(1-x)|<= |5/4 - x|
so we can square both sides:
1-x <= (5/4-x)^2= 25/16 - 10/4x +x^2 , x<=1
16-16x <= 25 -40x +16x^2 , x<=1
16x^2-24x+9>=0 , x<=1
(4x-3)^2>=0 , x<=1
x<=1


Your implication is the wrong way round I'm afraid.
Reply 221
Avatar for MEPS1996
MEPS1996
4
Original post by JosephML
Your implication is the wrong way round I'm afraid.

all steps are linked by a double implication arrow
Reply 222
Avatar for MEPS1996
MEPS1996
4
Original post by JosephML
Your implication is the wrong way round I'm afraid.

I know what your thinking. The squaring is non reversible, but since we are squaring the absolute value, it is reversible
Reply 223
Avatar for JosephML
JosephML
2
Original post by MEPS1996
all steps are linked by a double implication arrow


What you have shown is that if 1−x+x≤54\sqrt{1-x} + x \le \frac{5}{4} then 1−x+x\sqrt{1-x} + x is real, but this provides no information about whether the statement 1−x+x>54\sqrt{1-x} + x > \frac{5}{4} is true or not hence you haven't shown what was required.
Reply 224
Avatar for MEPS1996
MEPS1996
4
Original post by JosephML
Your implication is the wrong way round I'm afraid.

Thanks for your help anyway. Do you know how to plot the graph of y=sqrt(1-x^2)+sqrt(4-x^2)??
thanks
Reply 225
Avatar for MEPS1996
MEPS1996
4
Original post by JosephML
What you have shown is that if 1−x+x≤54\sqrt{1-x} + x \le \frac{5}{4} then 1−x+x\sqrt{1-x} + x is real, but this provides no information about whether the statement 1−x+x>54\sqrt{1-x} + x > \frac{5}{4} is true or not hence you haven't shown what was required.

I have also shown that if x+sqrt(1-x) are real, then x+sqrt(1-x) <= 5/4, since all steps are reversible.
Reply 226
Avatar for JosephML
JosephML
2
Original post by MEPS1996
I have also shown that if x+sqrt(1-x) are real, then x+sqrt(1-x) <= 5/4, since all steps are reversible.


Hm, I'm not convinced as you started by assuming your statement was true which is incorrect. You need to disprove 1−x+x>54\sqrt{1-x} + x > \frac{5}{4} i.e. by contradiction.
(edited 10 years ago)
Reply 227
Avatar for Noble.
Noble.
17
Original post by MEPS1996
I have also shown that if x+sqrt(1-x) are real, then x+sqrt(1-x) <= 5/4, since all steps are reversible.


As Joseph has pointed out, you've started with what you're trying to prove so the rest of your proof is useless (you've assumed it's true, what have you got left to show?). If this was an "if and only if" question the double implications would be fine (provided they're legit double implications that is) but answering an "If P then Q" question by starting with Q doesn't answer the question. All you really need to do though is rewrite the answers back to front.
Reply 228
Avatar for MEPS1996
MEPS1996
4
Original post by Noble.
As Joseph has pointed out, you've started with what you're trying to prove so the rest of your proof is useless (you've assumed it's true, what have you got left to show?). If this was an "if and only if" question the double implications would be fine (provided they're legit double implications that is) but answering an "If P then Q" question by starting with Q doesn't answer the question. All you really need to do though is rewrite the answers back to front.

You seem to think if id been asked to prove that x + sqrt(1-x) is real implies and is implied by x+sqrt(1-x) <= 5/4
then i would be correct. However, this statement is equivalent to x + sqrt(1-x) is real implies by x+sqrt(1-x) <= 5/4 AND x + sqrt(1-x) is implied by x+sqrt(1-x) <= 5/4. so i have indeed proved the former, which i what is needed.
Reply 229
Avatar for JosephML
JosephML
2
Original post by MEPS1996
You seem to think if id been asked to prove that x + sqrt(1-x) is real implies and is implied by x+sqrt(1-x) <= 5/4
then i would be correct. However, this statement is equivalent to x + sqrt(1-x) is real implies by x+sqrt(1-x) <= 5/4 AND x + sqrt(1-x) is implied by x+sqrt(1-x) <= 5/4. so i have indeed proved the former, which i what is needed.


Yes, but you start by assuming what you want to be true as true. What if it is false?!
Reply 230
Avatar for MEPS1996
MEPS1996
4
Original post by Noble.
As Joseph has pointed out, you've started with what you're trying to prove so the rest of your proof is useless (you've assumed it's true, what have you got left to show?). If this was an "if and only if" question the double implications would be fine (provided they're legit double implications that is) but answering an "If P then Q" question by starting with Q doesn't answer the question. All you really need to do though is rewrite the answers back to front.

Logicially, the process is the same as when you solve an equation, using all reversible steps. You know at the end the x values you got ARE the solutions to the equation. This is because you can make it true that x= the solutions
which implies that your original equation is true.
Reply 231
Avatar for MEPS1996
MEPS1996
4
Original post by JosephML
Yes, but you start by assuming what you want to be true as true. What if it is false?!

We know that x and sqrt(1-x) are real as we only allow x<=1, so it is true, which implies that original inequality true, (as we have reverse implication arrows)
Reply 232
Avatar for JosephML
JosephML
2
Original post by MEPS1996
Logicially, the process is the same as when you solve an equation, using all reversible steps. You know at the end the x values you got ARE the solutions to the equation. This is because you can make it true that x= the solutions
which implies that your original equation is true.


When you solve equations though, you are providing as a value for which they are true, they aren't themselves implicitly true.
Reply 233
Avatar for JosephML
JosephML
2
Original post by MEPS1996
We know that x and sqrt(1-x) are real as we only allow x<=1, so it is true, which implies that original inequality true, (as we have reverse implication arrows)


You seem to be missing the point, you can't start a proof with what you're trying to prove, it's just wrong.
Reply 234
Avatar for MEPS1996
MEPS1996
4
Original post by JosephML
You seem to be missing the point, you can't start a proof with what you're trying to prove, it's just wrong.

well just go through the steps in reverse, see any problems with it??
Reply 235
Avatar for Noble.
Noble.
17
Original post by MEPS1996
You seem to think if id been asked to prove that x + sqrt(1-x) is real implies and is implied by x+sqrt(1-x) <= 5/4
then i would be correct. However, this statement is equivalent to x + sqrt(1-x) is real implies by x+sqrt(1-x) <= 5/4 AND x + sqrt(1-x) is implied by x+sqrt(1-x) <= 5/4. so i have indeed proved the former, which i what is needed.


This is mostly a format and mathematical 'etiquette' issue, not a mathematical one, but any tutor would rip your answer to shreds, trust me :rofl:

You've been asked to show "if P, then Q" where P is xx and 1−x\sqrt{1-x} are real and Q is x+1−x≤54x + \sqrt{1-x} \leq \frac{5}{4} whereas your answer follows "If Q then P" which does not answer the question. By writing down x+1−x≤54x + \sqrt{1-x} \leq \frac{5}{4} you are assuming what you're trying to prove to be true.
Reply 236
Avatar for JosephML
JosephML
2
Original post by MEPS1996
well just go through the steps in reverse, see any problems with it??


Seems ok, would look a bit strange though.
Reply 237
Avatar for JosephML
JosephML
2
Original post by MEPS1996
Thanks for your help anyway. Do you know how to plot the graph of y=sqrt(1-x^2)+sqrt(4-x^2)??
thanks


Completely missed this, just do what you would normally do when sketching a graph. So look at dydx\frac{dy}{dx}, find the points where it meets the x and y axis (if it does at all) then connect the points.

Edit: Also, in this case you'll need to look at the values x can take such that y remains real.
(edited 10 years ago)
Reply 238
Avatar for MEPS1996
MEPS1996
4
Original post by Noble.
This is mostly a format and mathematical 'etiquette' issue, not a mathematical one, but any tutor would rip your answer to shreds, trust me :rofl:

You've been asked to show "if P, then Q" where P is xx and 1−x\sqrt{1-x} are real and Q is x+1−x≤54x + \sqrt{1-x} \leq \frac{5}{4} whereas your answer follows "If Q then P" which does not answer the question. By writing down x+1−x≤54x + \sqrt{1-x} \leq \frac{5}{4} you are assuming what you're trying to prove to be true.

Ok, thanks. So, within my answer, there is: if P then Q and if Q then P. so assuming P is true, since x<=1, this implies that Q is true. So to be more 'to the point' i should just write my proof in reverse with single headed implication arrows? this seems odd as joseph said, any ideas to clean it up?
Reply 239
Avatar for MEPS1996
MEPS1996
4
Original post by Noble.
This is mostly a format and mathematical 'etiquette' issue, not a mathematical one, but any tutor would rip your answer to shreds, trust me :rofl:

You've been asked to show "if P, then Q" where P is xx and 1−x\sqrt{1-x} are real and Q is x+1−x≤54x + \sqrt{1-x} \leq \frac{5}{4} whereas your answer follows "If Q then P" which does not answer the question. By writing down x+1−x≤54x + \sqrt{1-x} \leq \frac{5}{4} you are assuming what you're trying to prove to be true.

Ok, thanks. So, within my answer, there is: if P then Q and if Q then P. so assuming P is true, since x<=1, this implies that Q is true. So to be more 'to the point' i should just write my proof in reverse with single headed implication arrows? this seems odd as joseph said, any ideas to clean it up?

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