Proving an or statement in mathematical proofs

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In summary, the most common approach when proving a statement like "A implies B or C" is to assume A and that one of B or C is false, and then prove that A implies the remaining option (C or B). However, it is also common to split into cases and prove A implies both B and C separately. The approach to use depends on the problem at hand and there is no best general way.
  • #1
altcmdesc
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Proving an "or" statement

What is the general procedure when proving a statement like "A implies B or C"? Is it more common to assume A and then split into cases (i.e. case 1: A implies B, case 2: A implies C, therefore A implies B or C)? Or is it more common to assume A and that one of B or C is false and then prove that A implies C or B (depending on which is assumed false)?
 
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  • #2


Is it more common to assume A and then split into cases (i.e. case 1: A implies B, case 2: A implies C, therefore A implies B or C)? Or is it more common to assume A and that one of B or C is false and then prove that A implies C or B (depending on which is assumed false)?
Both are common. In my experience the most common is your second approach, i.e. assume A is true and B is false, and then conclude that C is true (interchange B and C if you want to). It's not uncommon either to do contradiction and assume A is true, B is false, C is false and then reach a contradiction, or possibly contraposition where you assume B and C are false and then conclude A is false. Really the approach to use depends on the problem and there is no best general way.
 
  • #3


If you did it the first way, you would have proven A implies B *and* C.
 
  • #4


Dragonfall said:
If you did it the first way, you would have proven A implies B *and* C.

I took his statement to mean:
Assume A
Show that either P or Q is true.
Case 1: Assume P is true. ... then B is true.
Case 2: Assume Q is true. ... then C is true.

which is valid.
 
  • #5


Where did P and Q come from?
 
  • #6


Dragonfall said:
Where did P and Q come from?

I introduced them to make the point more easily. When he wrote:
Case 1: A implies B
Case 2: A implies C
I'm reading it as a way of writing that in one case which I call P we prove that A implies B (we don't assume it), and in another case which I called Q we prove that A implies C. That is the only way I see for the argument to make sense.
 
  • #7


No, you still get A -> (B /\ C) with that approach.
 
  • #8


Dragonfall said:
No, you still get A -> (B /\ C) with that approach.

You must misunderstand me because I'm pretty sure my argument form is correct (whether altcmdesc meant it or not). For instance let:
A be "n is an integer square m^2".
B be "[itex]n \equiv 0 \pmod 3[/itex]".
C be "[itex]n \equiv 1 \pmod 3[/itex]"
Then:
Assume [itex]n=m^2[/itex] for some integer m.
Case 1 ([itex]m \equiv 1 \pmod 3[/itex] or [itex]m \equiv 2 \pmod 3[/itex]): Then [itex]n=m^2 \equiv 1 \pmod 3[/itex] which proves C in this case.
Case 2 ([itex]m \equiv 0 \pmod 3[/itex]): Then [itex]n=m^2 \equiv 0 \pmod 3[/itex] which proves B in this case.

This doesn't prove that every square is congruent to 0 AND 1 modulo 3, simply to one of them.

Here the conditions in parentheses are what I called P and Q and I think altcmdesc simply omitted them for simplicity. I think we're just understanding altcmdesc in different ways, so I guess we have to wait for him to clear up exactly what he meant.
 

1. What is the process for proving a statement?

The process for proving a statement involves using logical reasoning and evidence to demonstrate the truth of the statement. This may include conducting experiments, collecting data, or providing mathematical proofs.

2. Can a statement be considered proven without evidence?

No, in order for a statement to be considered proven, there must be evidence to support it. Without evidence, a statement remains unproven and is simply a hypothesis or belief.

3. How do you know when a statement has been proven?

A statement is considered proven when it has been rigorously tested and supported by evidence, and no counterexamples or contradictions have been found. Additionally, the statement must be able to withstand scrutiny and be consistent with existing knowledge and theories.

4. Is it possible to prove a statement to be 100% true?

In science, it is generally accepted that no statement can be proven to be 100% true. This is because new evidence or information may arise in the future that may challenge or alter our understanding of a statement. However, a statement can be considered highly probable or supported by a large body of evidence.

5. What is the difference between proving a statement and verifying a statement?

Proving a statement involves providing evidence and logical reasoning to demonstrate its truth, while verifying a statement involves confirming whether it is true or not. Verifying a statement may not necessarily involve providing evidence, but can be done through observation or simple tests. Proving a statement requires a more rigorous and systematic approach.

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