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Dialogue: 0,0:00:01.89,0:00:09.54,Default,,0000,0000,0000,,In this video we will look at a couple of examples to practice thinking about conjectures and counter examples.
Dialogue: 0,0:00:09.68,0:00:17.84,Default,,0000,0000,0000,,Example A says: here's an algebraic equation and a table of values for {\i1}n{\i0} with the results for {\i1}t{\i0}.
Dialogue: 0,0:00:18.04,0:00:20.45,Default,,0000,0000,0000,,Notice that we started out with this equation:
Dialogue: 0,0:00:25.56,0:00:28.72,Default,,0000,0000,0000,,And what we have here is really just a table,
Dialogue: 0,0:00:30.32,0:00:39.35,Default,,0000,0000,0000,,that's divided in three parts, we have our values for {\i1}n{\i0}, all sort of calculations in the middle, and our answer fot {\i1}t{\i0}.
Dialogue: 0,0:00:39.35,0:00:43.38,Default,,0000,0000,0000,,And after looking at the table, Pablo makes this conjecture:
Dialogue: 0,0:00:43.38,0:00:53.39,Default,,0000,0000,0000,,the value of ({\i1}n{\i0}-1)({\i1}n{\i0}-2)({\i1}n{\i0}-3), in other words, the answer for {\i1}t{\i0} is zero for any hole number value of {\i1}n{\i0}.
Dialogue: 0,0:00:53.39,0:00:58.94,Default,,0000,0000,0000,,So, he is basically saying, that no matter what number I plug in over on the left from {\i1}n{\i0},
Dialogue: 0,0:00:58.94,0:01:02.00,Default,,0000,0000,0000,,my answer is always going to end up being zero
Dialogue: 0,0:01:02.00,0:01:05.27,Default,,0000,0000,0000,,because that's what happen in the first three times. So maybe it will always happen.
Dialogue: 0,0:01:05.27,0:01:09.90,Default,,0000,0000,0000,,The question that we have is: is this a valid, or true, conjecture?
Dialogue: 0,0:01:09.90,0:01:14.81,Default,,0000,0000,0000,,So, if it's true, it means that it would be true for any number that you plug in for {\i1}n{\i0}.
Dialogue: 0,0:01:14.81,0:01:19.13,Default,,0000,0000,0000,,So you can plug in 100, and the answer should still be zero for {\i1}t{\i0}.
Dialogue: 0,0:01:19.13,0:01:25.50,Default,,0000,0000,0000,,So let's just test it out. Let's just try 100. Let's say {\i1}n{\i0}=100.
Dialogue: 0,0:01:25.50,0:01:31.08,Default,,0000,0000,0000,,We are trying to see if {\i1}t{\i0} will actually be equal zero.\NSo let's plug it in.
Dialogue: 0,0:01:31.32,0:01:39.12,Default,,0000,0000,0000,,We would have (100-1)(100-2)(100-3).
Dialogue: 0,0:01:39.44,0:01:41.28,Default,,0000,0000,0000,,100-1 is 99..
Dialogue: 0,0:01:41.68,0:01:43.66,Default,,0000,0000,0000,,Then we've got times 98,
Dialogue: 0,0:01:43.94,0:01:45.58,Default,,0000,0000,0000,,and times 97.
Dialogue: 0,0:01:45.95,0:01:49.19,Default,,0000,0000,0000,,Now I know that the answer to that is not zero,
Dialogue: 0,0:01:49.19,0:01:55.23,Default,,0000,0000,0000,,because to have an answer zero you have to multiply some of the one of the line by zero.
Dialogue: 0,0:01:55.23,0:02:01.53,Default,,0000,0000,0000,,So, this number is not equal zero. It's just gonna be a big number, definetly not equal to zero.
Dialogue: 0,0:02:01.71,0:02:06.37,Default,,0000,0000,0000,,So that means actually that his conjecture is not valid.\N
Dialogue: 0,0:02:07.14,0:02:15.40,Default,,0000,0000,0000,,It's not true and what I just did over here: {\i1}n{\i0} equals 100, that's a counter example.
Dialogue: 0,0:02:15.64,0:02:21.01,Default,,0000,0000,0000,,Because it's a specific example that shows that his conjecture is wrong.
Dialogue: 0,0:02:21.01,0:02:28.50,Default,,0000,0000,0000,,I can plug in, for example, 100 into that expression for {\i1}t{\i0} and my answer is not zero. Therefore, he is wrong.
Dialogue: 0,0:02:28.50,0:02:33.20,Default,,0000,0000,0000,,So a counter example is just one example to prove someone is wrong.
Dialogue: 0,0:02:33.20,0:02:38.81,Default,,0000,0000,0000,,That counter prefix means like going against the clean.
Dialogue: 0,0:02:39.02,0:02:42.07,Default,,0000,0000,0000,,Alright, let's look at example B:
Dialogue: 0,0:02:43.24,0:02:49.12,Default,,0000,0000,0000,,*Arthur is making figures for a graphic art project. He drew polygons and some of their diagonals.*
Dialogue: 0,0:02:49.28,0:02:55.71,Default,,0000,0000,0000,,And here we have four examples, based on these examples, Arthur made this conjecture:
Dialogue: 0,0:02:55.71,0:03:04.98,Default,,0000,0000,0000,,*if a convex polygon has {\i1}n{\i0} sides, then there are {\i1}n{\i0}-2 triangles drawn from any given vertex of the polygon.*
Dialogue: 0,0:03:04.98,0:03:10.52,Default,,0000,0000,0000,,So let's just think about what that means. He is saying that, if the shape has {\i1}n{\i0} sides,
Dialogue: 0,0:03:10.52,0:03:15.40,Default,,0000,0000,0000,,for example, this would be {\i1}n{\i0} equals 3, three sides,
Dialogue: 0,0:03:15.76,0:03:17.31,Default,,0000,0000,0000,,four sides,
Dialogue: 0,0:03:17.77,0:03:19.63,Default,,0000,0000,0000,,five sides,
Dialogue: 0,0:03:20.46,0:03:21.70,Default,,0000,0000,0000,,six sides.
Dialogue: 0,0:03:21.78,0:03:26.39,Default,,0000,0000,0000,,He is saying that there are always {\i1}n{\i0} minus 2 triangles.
Dialogue: 0,0:03:26.39,0:03:36.62,Default,,0000,0000,0000,,So, for example, if {\i1}n{\i0} is five, 5-2 is 3, he is saying, in this example, there are 3 triangles: one, two, three.
Dialogue: 0,0:03:36.62,0:03:42.91,Default,,0000,0000,0000,,And in the next one that are four triangles, which is 6-2.
Dialogue: 0,0:03:42.91,0:03:49.73,Default,,0000,0000,0000,,So, the question is: is Arthur's conjecture correct? Can you find a counter example to his conjecture?
Dialogue: 0,0:03:49.73,0:03:55.91,Default,,0000,0000,0000,,Well, his conjecture certainly appears to be correct based on the four examples that he has done.
Dialogue: 0,0:03:55.91,0:04:02.51,Default,,0000,0000,0000,,And we could do more examples and it would actually turn out that any example you'll try, it will be true,
Dialogue: 0,0:04:02.51,0:04:08.06,Default,,0000,0000,0000,,but you still haven't proven it, if you've just looked at examples, because it is still possible
Dialogue: 0,0:04:08.06,0:04:14.23,Default,,0000,0000,0000,,that that's another example out of there that you haven't thought out. That would be a counter example to his conjecture.
Dialogue: 0,0:04:14.23,0:04:15.97,Default,,0000,0000,0000,,So what we should say is:
Dialogue: 0,0:04:15.97,0:04:19.76,Default,,0000,0000,0000,,his conjecture appears to be true,
Dialogue: 0,0:04:22.41,0:04:24.50,Default,,0000,0000,0000,,but it still needs to be proven.
Dialogue: 0,0:04:26.69,0:04:35.40,Default,,0000,0000,0000,,Because just looking at examples isn't an official really proof for sure that it will always be true.