# 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

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?

**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?

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

(Original post by

okay, so i should be thinking:

P implies Q is false because P is sufficient for Q but Q is not necessary for P

**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

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

(Original post by

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?

**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?

**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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LAMP

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#7

(Original post by

No. You should be deciding

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.

**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.

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rnd

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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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#9

(Original post by

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.

**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.

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generalebriety

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#10

(Original post by

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?

**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?

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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

(Original post by

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

**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

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