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
00:25not all your marks. So
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.