00:01
We want to find a counter example, if possible, to the following universally quantified statement where the universe for each variable consists of all integer numbers.
00:17
Statement is, for all x exists y, such that 2x plus 5y is less than equal to x plus y.
00:26
So let's suppose we can find a counter example.
00:29
That is, if this is not true, if we find a counter example, it means that this statement is not true.
00:41
And if this statement is not true, the negation of the statement is true.
00:50
That is, it's the statement, the statement is not true means it is false, and so its negation is true.
00:57
So what's the negation of this statement here? there exists x such that for all y it happens that to x plus 5y is greater than x plus y okay so the negation of this statement is this one so if you find a contrary example for the statement given here means the statement is false and then its negation is true so what happens what what happened if this statement is true means there exists an integer number because we are the universe for x and y is the real the integer numbers to exist an integer number such that for all integer numbers it happens that 2x plus 5y greater than x plus y that's the same as there exists an integer number such that for all y into the number it happens that then we can arrange this inequality this way to x minus x greater than y minus 5 y that's the same as there exists an integer number x such that for all y into the number we have x greater than negative for y or what is the same there exist an integer number such that for all z for a while sorry in this in the numbers we have y in this case we are going to divide by negative four so it's going to change the inequality so y greater than uh negative x over four let's verify that if you want x is squared than negative 4 y we divide by negative 4 and that means this inequality changes since nette 4 is a negative number and that means x over negative 4 less than y is the same as this inequality here and so what it means what this means is that we have found in this statement true this one here this one here being true means that there exist an integer number such that for all integer numbers y, y is greater than negative x over 4.
04:11
What this means is that negative x over 4 is a lower bound for the integer for the integer numbers because for all y into the number, y must be created than this number.
04:26
So this is equivalent to say that negative x that's the x that exists is an integer number.
04:33
And the value negative x over 4 is a lower bound for the sets for the set sorry of integer numbers.
04:51
But this is not true.
04:53
The set of interest numbers is not bounded below.
04:57
It's not bounded below nor above.
05:01
So this is this is not true because the set of integer numbers is not bounded.
05:28
Above or below so it's not bound to the row in any sense so this statement is not true so the statement exists x such that for all y i'm not going to put the set because we know it's in the numbers so so that for y it happens that to x to x plus y greater 2x plus 5 y is greater than x plus y is false cannot be true if we consider it to be true we find a contradiction with the fact that the set of integer number is not bounded the statement is equivalent to say that the set of intial numbers have has a bound negative x over so the statement is false, but this statement being this statement, the denigation of the given the given statement that is for x, there exists y, such as 2x plus 5y less than or equal to x plus 5, we have that this one that is the given statement let me put it again it doesn't matter if i repeat it very clear so we have that the given statement how it is well negation the negation here is false okay it means the original statement is true so we cannot find a counter example.
08:28
So we summarize we suppose that we can find a counter example to the given statement it means the statement is false, so his negation is true and that leads to contradiction.
08:40
So the statement cannot we cannot find a counter example to the given statement so the statement is true.
08:48
And that's the conclusion.
08:51
The given statement is true.
08:52
This statement here is true.
08:56
Then let's analyze what this statement says.
08:59
That the statement says that the says that for all x, there exists a number for all integer x exists another integer y, such that 2x plus 5y is less than or equal to x plus y.
09:13
So let's work with this inequality as we did before.
09:17
So this is equivalent to say that for all x integer x exists another integer y, such that 2x minus x less than or equal to y minus 5y.
09:32
That's equivalent to for all integer x exist an integer z a y such that x is less than or equal to negative four y.
09:46
And the y we got to find, we get to choose given x is then if we divide by negative four, we get negative x over four greater than or equal.
10:05
So if we divide by negative four, the inequality changes sense...