What does this statement even mean?

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  • #1
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Hi,

I'm not sure this should go in homework of here as this was a test question, but the question its self isn't a test question.

I got this question marked wrong, for the record.

Question
For each statement below, determine if the statement is true or false. If true, provide a proof, if false, provide a counterexample.


A.) If A is a nonzero matrix, and if AB = 0 (the zero matrix), then B = 0

Suppose [tex]A[/tex] has a inverse and[tex] B ≠ 0 [/tex]

Then,

[tex] A^{-1}AB = 0{A^-1}[/tex]
[tex]I_nB = 0[/tex]


Since Identity * B ≠ 0 unless B is the zero matrix, B must be zero.


However, the professor simply wrote false and then gave an example of how it was possible for the matrix to not be the zero matrix.


However, I clearly showed B can be zero.

So, the real question I'm asking is if this question was to be seen in any sense what is the context of the equal sign; in terms of any mathematics to make a statement false of true.

For example would this statement also be false..in terms of the question below.
For each statement below, determine if the statement is true or false. If true, provide a proof, if false, provide a counterexample.


Let f(x) = 4 and f(x) = x^2, then x = -2.


I would have to say false?

Because,

Let f(x) = 4 and f(x) = x^2, then x = -2 and x = 2.



Comments please, and rip it apart :D
 

Answers and Replies

  • #2
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Quick summary,

What makes this statement false?

A.) If A is a nonzero matrix, and if AB = 0 (the zero matrix), then B = 0
 
  • #3
Deveno
Science Advisor
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[0 1][0 1]....[0 0]
[0 0][0 0] = [0 0]

so
 
  • #4
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[0 1][0 1]....[0 0]
[0 0][0 0] = [0 0]

so
Explain, please.

:)
 
  • #5
Deveno
Science Advisor
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is my matrix A non-zero? yes.

is the product AB 0? yes.

does B = 0? no.

counter-example win.
 
  • #6
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But how does that make the statement false? does the = sign mean must be for fall cases?
 
  • #7
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I showed example of a case where B does equal 0 and you showed a case of where it isn't zero.

How is the statement

"If A is a nonzero matrix, and if AB = 0 (the zero matrix), then B = 0 "

false automatically?
 
  • #8
Deveno
Science Advisor
906
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to make a statement true, it must be true for ALL A and B satisfying the assumptions.

to make a statement false, it just needs to be false for ONE A and B satisfying the assumptions.

the statement:

if n is an integer, n > 6

is false, even though some integers are greater than 6.
 
  • #9
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So, = sign always means "if and only if" ?
 
  • #10
Deveno
Science Advisor
906
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which "=" sign are you talking about?

you have a statement of the form:

If _____ and ______, then ______.

if there is any instance where the first two blanks are true, and the third blank is false, the statement is false. it only takes ONE counter-example to disprove something, where to prove something, it has to be true for ALL examples.

that's why theorems are so powerful. a statement like 3+5 = 5+3 might be a coincidence. a statement like A+B = B+A, for all A and B, is a sweeping statement, which requires proof.
 
  • #11
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Yea I got that part; but, still don't understand how you get

If A is a nonzero matrix, and if AB = 0 (the zero matrix), then B = 0

is the same statement as

If A is a nonzero matrix, and if AB = 0 (the zero matrix), then B must be 0

:(
 
  • #12
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The "=" is not what makes it mean "for all A and B."

"If ___ then ___" constructions mean that the "then" part is always true as soon as the if part is true. One implies the other.

As a general rule in math, if he wanted to know if it was possible, then he would've used that terminology. Essentially, everything is always as general as it possible can be by default, and as we add onto the sentence, it narrows the scope, but never the other way around.
 
  • #13
Deveno
Science Advisor
906
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well, as i demonstrated earlier, it is possible that A ≠ 0, and AB = 0, but B ≠ 0.

i understand that you're thinking: "well, sure, but it might be true that B = 0 anyway".

and that part is true, but you're missing the "if...then..." bit.

the sentence is asserting that if certain conditions hold, a certain conclusion holds.

we can't say that just because A ≠ 0, and AB = 0, that B = 0.

it's not a matter of "if and only if", it's just that the conditions A ≠ 0, AB = 0,

aren't "enough" to say that B = 0.

this doesn't mean B can't be 0, we just can't tell.
 
  • #14
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well, as i demonstrated earlier, it is possible that A ≠ 0, and AB = 0, but B ≠ 0.

i understand that you're thinking: "well, sure, but it might be true that B = 0 anyway".

and that part is true, but you're missing the "if...then..." bit.

the sentence is asserting that if certain conditions hold, a certain conclusion holds.

we can't say that just because A ≠ 0, and AB = 0, that B = 0.

it's not a matter of "if and only if", it's just that the conditions A ≠ 0, AB = 0,

aren't "enough" to say that B = 0.

this doesn't mean B can't be 0, we just can't tell.
*click*


Thanks so much :P
 
  • #15
HallsofIvy
Science Advisor
Homework Helper
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Hi,

I'm not sure this should go in homework of here as this was a test question, but the question its self isn't a test question.

I got this question marked wrong, for the record.

Question
For each statement below, determine if the statement is true or false. If true, provide a proof, if false, provide a counterexample.


A.) If A is a nonzero matrix, and if AB = 0 (the zero matrix), then B = 0

Suppose [tex]A[/tex] has a inverse and[tex] B ≠ 0 [/tex]

Then,

[tex] A^{-1}AB = 0{A^-1}[/tex]
[tex]I_nB = 0[/tex]


Since Identity * B ≠ 0 unless B is the zero matrix, B must be zero.


However, the professor simply wrote false and then gave an example of how it was possible for the matrix to not be the zero matrix.


However, I clearly showed B can be zero.
So what? The question did not ask if B could be 0, it asked if "B= 0" was a necessary conclusion.

So, the real question I'm asking is if this question was to be seen in any sense what is the context of the equal sign; in terms of any mathematics to make a statement false of true.
For example would this statement also be false..in terms of the question below.
For each statement below, determine if the statement is true or false. If true, provide a proof, if false, provide a counterexample.


Let f(x) = 4 and f(x) = x^2, then x = -2.


I would have to say false?
Yes. The statement "if P then Q" is true only if, any time P is true, Q is also true. Here, it is possible that x= 2 so that "P" (x^2 =4) is true but "Q" (x= -2) is false.

Because,

Let f(x) = 4 and f(x) = x^2, then x = -2 and x = 2.
Actually, that's a nonsense statement: the conclusion "x= -2 and x= 2" is impossible. x cannot take on two different values. If the conclusion were "then x= -2 or x= 2" then it would be true.


Comments please, and rip it apart :D
 

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