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necessary and sufficient conditions.
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LAMP
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#1
I;d appreciate if someone could check my thinking is correct here.
For integers n: 4 divides n is a ......... condition for 4 divides n^2
P = 4 divides n
Q = 4 divides n^2
P is false, counter - example: 4/5
Q is also false = 4/(3^2)
Therefore P implies Q and Q implies P is true and so the answer is "necessary and sufficient"
For integers n: 4 divides n is a ......... condition for 4 divides n^2
P = 4 divides n
Q = 4 divides n^2
P is false, counter - example: 4/5
Q is also false = 4/(3^2)
Therefore P implies Q and Q implies P is true and so the answer is "necessary and sufficient"
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generalebriety
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#2
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#2
No, you've got the idea completely wrong.
If P implies Q, then P is a sufficient condition for Q (and Q is necessary for P). If Q implies P, then P is a necessary condition for Q (and Q is sufficient for P). Can you see why?
Which way round is it here?
If P implies Q, then P is a sufficient condition for Q (and Q is necessary for P). If Q implies P, then P is a necessary condition for Q (and Q is sufficient for P). Can you see why?
Which way round is it here?
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rnd
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#3
LAMP
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#4
(Original post by generalebriety)
No, you've got the idea completely wrong.
If P implies Q, then P is a sufficient condition for Q (and Q is necessary for P). If Q implies P, then P is a necessary condition for Q (and Q is sufficient for P). Can you see why?
Which way round is it here?
No, you've got the idea completely wrong.
If P implies Q, then P is a sufficient condition for Q (and Q is necessary for P). If Q implies P, then P is a necessary condition for Q (and Q is sufficient for P). Can you see why?
Which way round is it here?
okay, so i should be thinking:
P implies Q is false because P is sufficient for Q but Q is not necessary for P
in the same way, Q implies P is false because Q is sufficient for P but P is not necessary for Q?
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generalebriety
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#5
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#5
(Original post by LAMP)
okay, so i should be thinking:
P implies Q is false because P is sufficient for Q but Q is not necessary for P
okay, so i should be thinking:
P implies Q is false because P is sufficient for Q but Q is not necessary for P
Think about it. Does the statement "4 divides n" imply the statement "4 divides n^2"? Does the statement "4 divides n^2" imply the statement "4 divides n"? If your answer to either of these is yes, prove it; if your answer to either is no, give a counterexample.
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DFranklin
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#6
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#6
(Original post by LAMP)
okay, so i should be thinking:
P implies Q is false because P is sufficient for Q but Q is not necessary for P
in the same way, Q implies P is false because Q is sufficient for P but P is not necessary for Q?
okay, so i should be thinking:
P implies Q is false because P is sufficient for Q but Q is not necessary for P
in the same way, Q implies P is false because Q is sufficient for P but P is not necessary for Q?
In other words, suppose we know 4 divides n. Does it follow that 4 must divide n^2? If so, P implies Q. If not, it doesn't.
Conversely, suppose we know 4 divides n^2. Does it follow that 4 must divide n? If so, Q implies P. If not, it doesn't.
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LAMP
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#7
(Original post by DFranklin)
No. You should be deciding from the specific question about divisibility whether P implies Q, and whether Q implies P.
In other words, suppose we know 4 divides n. Does it follow that 4 must divide n^2? If so, P implies Q. If not, it doesn't.
Conversely, suppose we know 4 divides n^2. Does it follow that 4 must divide n? If so, Q implies P. If not, it doesn't.
No. You should be deciding from the specific question about divisibility whether P implies Q, and whether Q implies P.
In other words, suppose we know 4 divides n. Does it follow that 4 must divide n^2? If so, P implies Q. If not, it doesn't.
Conversely, suppose we know 4 divides n^2. Does it follow that 4 must divide n? If so, Q implies P. If not, it doesn't.
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rnd
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#8
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#8
if 4|n (say n/4= k and k is an integer)
then
n^2/4 = kn another integer. i.e. If 4|n then 4|n^2
However 4|n^2 does not imply 4|n since 4|2^2 but 4 does not divide 2.
I hope I haven't said too much.
then
n^2/4 = kn another integer. i.e. If 4|n then 4|n^2
However 4|n^2 does not imply 4|n since 4|2^2 but 4 does not divide 2.
I hope I haven't said too much.
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LAMP
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#9
(Original post by MAR)
if 4|n (say n/4= k and k is an integer)
then
n^2/4 = kn another integer. i.e. If 4|n then 4|n^2
However 4|n^2 does not imply 4|n since 4|2^2 but 4 does not divide 2.
I hope I haven't said too much.
if 4|n (say n/4= k and k is an integer)
then
n^2/4 = kn another integer. i.e. If 4|n then 4|n^2
However 4|n^2 does not imply 4|n since 4|2^2 but 4 does not divide 2.
I hope I haven't said too much.
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generalebriety
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#10
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#10
(Original post by LAMP)
So what you;ve said is that if n/4 gives an integer answer then n^2/4 will always give an integer answer but not the other way around?
So what you;ve said is that if n/4 gives an integer answer then n^2/4 will always give an integer answer but not the other way around?
If 4 divides n, then n/4 is an integer. Does n/4 being an integer imply that n^2/4 is an integer? Does n^2/4 being an integer imply n/4 is an integer?
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Blue Bumble Bee
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#11
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#11
We have:
P = 4 divides n
Q = 4 divides n^2
and we want to find a way to express:
- if 4 divides n then 4 divides n^2
we know that p => Q as whenever 4 divides n, 4 will also divide n^2.
proving this goes as follows:
if 4 divides n then n = 4k
therefore n^2 =16k^2
therefore n^2 = 4(4k^2)
so, n^2 has a factor of 4
as P implies Q the statement is sufficient.
-----------------------------------------------------------
to check necessity lets assume that Q implies P
so, if 4 is a factor of n^2 then 4 is a factor of n
lets take 2 for example as this provides a counter example
2^2 is a factor of 4 however, 4 is not a factor of 2,
-----------------------------------------------------------
therefore the statement is sufficient but not necessary
P = 4 divides n
Q = 4 divides n^2
and we want to find a way to express:
- if 4 divides n then 4 divides n^2
we know that p => Q as whenever 4 divides n, 4 will also divide n^2.
proving this goes as follows:
if 4 divides n then n = 4k
therefore n^2 =16k^2
therefore n^2 = 4(4k^2)
so, n^2 has a factor of 4
as P implies Q the statement is sufficient.
-----------------------------------------------------------
to check necessity lets assume that Q implies P
so, if 4 is a factor of n^2 then 4 is a factor of n
lets take 2 for example as this provides a counter example
2^2 is a factor of 4 however, 4 is not a factor of 2,
-----------------------------------------------------------
therefore the statement is sufficient but not necessary
Last edited by Blue Bumble Bee; 9 months ago
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Sodium229
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#12
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#12
(Original post by Blue Bumble Bee)
We have:P = 4 divides nQ = 4 divides n^2and we want to find a way to express: - if 4 divides n then 4 divides n^2we know that p => Q as whenever 4 divides n, 4 will also divide n^2.proving this goes as follows:if 4 divides n then n = 4ktherefore n^2 =16k^2therefore n^2 = 4(4k^2)so, n^2 has a factor of 4as P implies Q the statement is sufficient.-----------------------------------------------------------to check necessity lets assume that Q implies Pso, if 4 is a factor of n^2 then 4 is a factor of nlets take 2 for example as this provides a counter example2^2 is a factor of 4 however, 4 is not a factor of 2, -----------------------------------------------------------therefore the statement is sufficient but not necessary
We have:P = 4 divides nQ = 4 divides n^2and we want to find a way to express: - if 4 divides n then 4 divides n^2we know that p => Q as whenever 4 divides n, 4 will also divide n^2.proving this goes as follows:if 4 divides n then n = 4ktherefore n^2 =16k^2therefore n^2 = 4(4k^2)so, n^2 has a factor of 4as P implies Q the statement is sufficient.-----------------------------------------------------------to check necessity lets assume that Q implies Pso, if 4 is a factor of n^2 then 4 is a factor of nlets take 2 for example as this provides a counter example2^2 is a factor of 4 however, 4 is not a factor of 2, -----------------------------------------------------------therefore the statement is sufficient but not necessary
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