AASL 1.6 Simple deductive proofs
Learn AASL 1.6 Simple deductive proofs in this free IB Mathematics Analysis and Approaches (AA) video lesson for SL 1.6—Simple proof.
Learn AASL 1.6 Simple deductive proofs in this free IB Mathematics Analysis and Approaches (AA) video lesson for SL 1.6—Simple proof.
Learn AASL 1.6 Simple deductive proofs in this free IB Mathematics Analysis and Approaches (AA) video lesson for SL 1.6—Simple proof.
The video lesson focuses on deductive proofs, emphasizing the importance of following a structured method to avoid losing marks. It covers both numerical and algebraic proofs, illustrating how to manipulate the left-hand side (LHS) to match the right-hand side (RHS) or vice versa, without crossing the dividing line between them.
Key points include:
00:00Hi everybody, so in this
00:01lesson we're going to do
00:02some simple deductive proofs now.
00:04It says simple there, but
00:06these are not always simple.
00:10So these are the only
00:11formal proofs we have to
00:13do at standard level. And
00:15they're not that formal, but
00:17we do have a method
00:18that we need to set
00:18out correctly. And if you
00:20don't do the correct steps,
00:23you will lose some if
00:25
00:26what we do is
00:28Well, we have to prove
00:29both numerical proofs like this
00:33one, which is of numbers,
00:35and algebraic proofs like this
00:36where we've got some letters
00:37in there. And we have
00:39an equal sign, and this
00:40isn't actually a typo. This
00:43is the identity symbol, which
00:45I'll get to in a
00:46second. So what we're going
00:47to do is we have
00:48a left hand side, which
00:49we can just call LHS,
00:51then I'm going to put
00:52a line straight down the
00:53middle, and I have a
00:55right hand side.
00:56And what we effectively have
00:58to do is make the
01:00left hand side look like
01:01the right hand side or
01:03the right hand side look
01:04like the left hand side
01:05doesn't matter which way you
01:06go. So the left hand
01:08side is two thirds plus
01:10one twelfth. Let me just
01:14move that over here. Two
01:15thirds plus one twelfth and
01:18the right hand side is
01:19three quarters. So obviously this
01:24is that it's better to
01:25choose the left hand side
01:26and make it look like
01:27the right hand side. Now
01:28I can't, you're not allowed
01:30cross this line let's say.
01:32So I can't subtract three
01:33quarters and make it equal
01:34to zero or anything like
01:35that. I can't solve it
01:36like like an equation. I
01:38need to make this look
01:39like this or the other
01:41way around. And each step
01:43has to, it has to
01:45follow the previous step logically
01:47and make sense. And that's
01:48why it's a proof. A
01:51rigorous proof.
01:52So two -thirds plus one
01:53-twelfth. I'm going to make
01:55the denominator common. So I
01:59need a common denominator. I'm
02:00going to choose twelfth. So
02:02this is plus twelfth. Two
02:04-thirds in twelfth is eight
02:06-twelfth plus one -twelfth equals
02:09nine -twelfth, nine -twelfth, which
02:13equals three -quarters. And then
02:16at the end, I'm going
02:17to write left. So this
02:19is my left -hand side.
02:20You can clearly see my
02:22left hand side equals right
02:24hand side. I don't have
02:25to keep writing three quarters
02:26or anything like that. I'm
02:27just going to write left
02:29hand side equals right hand
02:32side and then I'm going
02:34to write QED. QED stands
02:37for Quad -era Demonstrandum, which
02:40is Latin for which was
02:43to be demonstrated. Quad -era
02:46Demonstrandum. So just write
02:48If I read you a
02:50proof, you just write QED,
02:51it's like there. It's like
02:52saying there, I've proved it
02:53for you. Now, next question.
02:56So we're going to do
02:57the same for these. Now,
02:59the identity symbol, so it's
03:02very similar to the to
03:04the equals sign, but the
03:06identity symbol means that the
03:08left hand side equals the
03:11right hand side for all
03:13values of x. So for
03:14example, look x plus
03:16three equals five. That's an
03:20equation. And the answer is
03:22x equals two. But I
03:23can't write x plus three
03:25is equivalent to five because
03:30it's not. If x was
03:33anything other than two, then
03:35these are not the same
03:36thing. So that's just the
03:39identity symbol or the equivalence
03:41symbol. So it's like saying
03:42this is equivalent to this.
03:44I'm going to do the
03:45same thing when I'm going
03:45to have left hand side,
03:46left hand side, and right
03:50hand side. Now again, do
03:54I want to turn the
03:55left one into the right
03:56one or the right one
03:57into the left one? Well,
03:58you could complete the square
04:02here if you want it.
04:03But again, I think it's
04:05definitely easier if I just
04:07multiply this out. So this
04:08is x minus 2 squared
04:11plus 7.
04:12I multiply this out and
04:14make it look like that.
04:14I think that's definitely easier.
04:16So I'm going to make
04:16this equal x squared minus
04:214x plus 4 plus 7.
04:26OK, if you think I've
04:27done that too quickly in
04:29my head at the side,
04:31do it the long way,
04:32x minus 2 times x
04:34minus 2. This times this,
04:36this times this, this times
04:38this.
04:40I would highly recommend you
04:42just learn how to do
04:43that though. A plus B
04:45squared is A squared plus
04:462A B plus B squared.
04:48Anyway, that equals x squared
04:51minus 4x plus 4 plus
04:5611, sorry plus 4 plus
04:577 is plus 11, which
05:00equals the right hand side.
05:02So I can therefore again
05:03say left hand side equals
05:06right hand side, Q,
05:08E D done. Last one.
05:14So when I said, when
05:16I said that these weren't
05:17always simple, here's a prime,
05:21perfect example of that. So
05:22I'm going to do the
05:23same thing, left hand side,
05:25right hand side, a line
05:29straight down. I don't know
05:32if I need that much
05:33space. Left hand side is
05:34one over M. So this
05:35is the situation where
05:36I am better off. It's
05:40going to be easier for
05:41me to make the right
05:43look like the left than
05:45to make the left look
05:46like the right. So I'm
05:47going to make I am
05:49going to try and simplify
05:50this. So it's 1 over
05:52m plus 1 plus 1
05:54over m squared plus m.
05:58Okay. These are algebraic fractions.
06:01I need to I need
06:03a I need
06:04a common denominator. So firstly,
06:09let me just make it
06:10look like this. If I
06:12factorize the denominator here, I
06:14have m into m plus
06:161. Okay, so the denominator
06:20here is m plus 1,
06:21and the denominator here is
06:23m times m plus 1.
06:24What denominator can I choose
06:26that this goes into and
06:27this goes into? Well, the
06:29answer is this.
06:33Because m plus 1 goes
06:36into this and m times
06:38m plus 1 goes into
06:39this obviously. So this is
06:41my denominator. If you want
06:42to think of that, what
06:44I've just done numerically, imagine
06:47I said what's the common
06:48denominator between 1 over 5
06:51and 1 over 5 times
06:567, or the common denominator,
07:01of V, five times seven,
07:03or 35, because five goes
07:04into it. Okay, let me
07:06get rid of that. So,
07:11this is obviously just one,
07:12because it's the exact same
07:12thing, but this thing, I've
07:14times the denominator by M,
07:16so I need to times
07:16the numerator by M, one
07:18times M, is M. Okay,
07:20get in there. This now
07:22equals, the denominator is the
07:25same, M times M plus
07:27one.
07:29So I can just add
07:30the numerators, it's m plus
07:321, m plus 1, these
07:35cancel, the m plus 1
07:37cancel. So this equals 1,
07:42and cancel them there if
07:43I want, equals 1 over
07:46m. Left hand side equals
07:50right hand side, one over
07:52m, one over m, q,
07:54e, d. If you don't
07:56write q,
07:57D or left and side
07:58equals right hand side. Yeah,
08:00you lose a mark because
08:01you haven't formally proved it.
08:04You have to show all
08:06these steps. So you go,
08:07you do a left hand
08:08side, right hand side, decide
08:10which one you want to
08:11turn into the other one.
08:13You can't cross the line,
08:15as I said before, add
08:16and subtract across the line.
08:18And at the end, when
08:19they're equal, you say left
08:20hand side equals right hand
08:20side and QED.