Logic: if A leads to B, then if I have A^ (not A) then is it going to satisfy what's claimed?

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Ryan$

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Hi guys, I'm totally confused about dilemma on mathematics ; if A leads to B, then if I have A^ (not A) then is it going to satisfy what's claimed?!
for example lets say I have array and if I choose one element of the array, then that element must satisfy every element on the array in its left is smaller than it, and the right of that element is bigger than it, for example : {1 , 2 , 3 } if I choose the second index which its value 2, then yeah the array would be {1,2,3} , all element on the right of 2 is bigger than it and the left of it is smaller than it !
but if I choose the first index which it's 1, then there's no elements on its left, so how can I solve the question?! it's like (not A) then what I'm doing ... is it by default satisfy the required on the question?! thanks alot.
 
I have no requirements / data on what will happen if I don't have numbers on the right/left of the chosen element ....so what math says on that dilemma ?
 
Please give the exact statement of the problem you are trying to solve. When you jump into the middle of something, it's hard to follow your thinking.

It sounds like you are trying to prove something about a sequence, such as that it is increasing, and perhaps are being confused by the concept of vacuous truth. That is, if a sequence is defined as increasing if each element is greater than all elements on its left, then when applied to the first element, there are no elements on the left, so we consider it true. This is a convention of mathematical language: any statement about all elements of a set is considered to be true when the set is empty, because there is no element for which it is not true.

Is that something like what you are asking?
 
Dr.Peterson said:
Please give the exact statement of the problem you are trying to solve. When you jump into the middle of something, it's hard to follow your thinking.

It sounds like you are trying to prove something about a sequence, such as that it is increasing, and perhaps are being confused by the concept of vacuous truth. That is, if a sequence is defined as increasing if each element is greater than all elements on its left, then when applied to the first element, there are no elements on the left, so we consider it true. This is a convention of mathematical language: any statement about all elements of a set is considered to be true when the set is empty, because there is no element for which it is not true.

Is that something like what you are asking?
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Yup exactly! but what do you mean exactly vacuous true?
the last statement of "because there's no element for which it's not true" still no understand able for me, maybe please illustrate it? can you give me an example to illustrate it more?
if I don't know anything about something, so how I claim that's true?!
 
Ryan$ said:
Yup exactly! but what do you mean exactly vacuous true?
the last statement of "because there's no element for which it's not true" still no understand able for me, maybe please illustrate it? can you give me an example to illustrate it more?
if I don't know anything about something, so how I claim that's true?!
Click to expand...
Hi Rayn$, Do you understand why \(\displaystyle \bf{\text{ if }t\in A\text{ and }t\notin A \text{ then }\pi<3}\) is a true statement for all \(\displaystyle \bf{t}~?\).
 
Did you read the Wikipedia link I attached to the term vacuous truth? That was there so you could learn what I meant. (I also used the term so you could search for it and find other explanations.)
 
pka said:
Hi Rayn$, Do you understand why \(\displaystyle \bf{\text{ if }t\in A\text{ and }t\notin A \text{ then }\pi<3}\) is a true statement for all \(\displaystyle \bf{t}~?\).
Click to expand...
Ryan$ said:
What's A?!
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Do you understand that \(\displaystyle \bf{t\in A\text{ and }t\notin A}\) is always false no matter what A is or what t is?

You brought this up. You wrote that you had (A and not A) That is a false statement no matter what A is. It makes no difference.
"(A and not A) implies X" is a true statement no matter what X is.
It is always true that A false statement implies any statement.
It is always true that A true statement is implied by any statement.

You need to do a course in formal logic.
 
Ryan$ said:
I'm totally confused about dilemma on mathematics ; if A leads to B, then if I have A^ (not A) then is it going to satisfy what's claimed?!
Click to expand...
This is the part of your question that pka is dealing with; I pretty much ignored it, because it doesn't seem to fit with the rest, and I don't know what you have in mind here.

You still haven't answered my question about the specific problem you are working on; telling us that would give us the necessary context to be able to answer you appropriately. Specific examples are much easier to explain than vague generalities.
 
An "implication" is a statement that "If A then B". That means that if statement A is true then statement B must also be true. It says nothing at all about what happens if A is not true!

A teacher might say "If you get every problem on this test correct then you will get an "A" for the test". One would certainly expect that to be true! Suppose you actually got one problem wrong but all the others correct. If the teacher gives you an A anyway, does that mean he/she lied? NO! The teacher said nothing about what would happen if you did NOT get all the questions correct.
 
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