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When you are given information and asked to show 'something' is it completely wrong to start with that 'something' and work back to the information?

Can you start with what you want to show, work backwards a few steps using equivalence and then show that.For example I was told i cant work backwards in the problem:

Let $a,b≥2$ Show that $a+b≤ab$. What i did which was wrong was start with what i had to show then reached $(a-1)(b-1)≥1$ which i said was satisified by $a,b≥2$. This is backwards and wrong,

I know you should start with $a+b≤ab$ show equivalence to $a+b-ab≤0$ and then i think the problem reduces to showing $a+b-ab$ is greater than 0 but its quite confusing to me if i am working backwards here or not because if you start from $a,b>2$ and go to $a+b-ab$ it seems as though you have.

Thanks

Carlos Bacca
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    $B\implies A$ does not imply that $A\implies B$. – Angina Seng Oct 28 '18 at 13:21
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    It's not always a bad idea...but in order to get a solid proof you'll need to verify that each of your steps is reversible. Even if you can't do that, it's possible that the "backwards" argument will generate some good ideas which you can use. – lulu Oct 28 '18 at 13:24

2 Answers2

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To show that a certain statement $P$ is true, you can work backwards. However, you need to make sure that every step is reversible, meaning, if $P\implies S$, then $S\implies P$ as well. In your example, you have $$a+b\leq ab \iff ab-a-b+1\geq1 \iff (a-1)(b-1)\geq 1$$ which is true for the given conditions. Now you can start from the last statement, which is true, and go in the reverse direction as permitted by the direction of the arrows to reach the statement you are trying to prove. This way is valid. However, if you only had $$a+b\leq ab \implies ab-a-b+1\geq1 \implies (a-1)(b-1)\geq 1$$ where at least one arrow only points in the forward direction, then even if you reach a correct conclusion, you cannot say that the initial statement was true. This is because your logic doesn't work the other way around anymore. For example, $$0\geq1 \implies 0\cdot0\geq0\cdot1 \iff 0\geq0$$ which is true, but since the first arrow only points forward, the initial statement need not be true (indeed, it's false in this case).

YiFan Tey
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  • Thanks for the answer, but given the question, is your first argument (in the forward direction) not a backwards argument, thats what i was told. i was told that i should start with a,b>=2 – Carlos Bacca Oct 28 '18 at 14:29
  • Well, whoever told you that is wrong ;) – YiFan Tey Oct 28 '18 at 14:29
  • i got 0 marks for that – Carlos Bacca Oct 28 '18 at 14:31
  • I suspect your issue is that, as in the second argument in my answer, you demonstrated that $a+b\leq ab\implies(a-1)(b-1)\geq1$, and stated that this is true, then went on to say that the initial statement is true as well. This is a wrong argument. However, if you make it crystal clear that every step along the way is reversible and is an "if and only if" statement, the argument is perfectly okay and your teacher has no reason to mark you wrong. – YiFan Tey Oct 28 '18 at 14:39
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As far as a method to generate a proof, working backwards a bit can provide some insight into what needs to happen when you write your proof. As for why you can't reverse the implication (go backwards)

If it's raining, then it is wet outside.

is not the same as

If it is wet outside, then it's raining.

RhythmInk
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