Induction concluding statements (1 Viewer)

[cutemouse]

Hi,

I recently read some stuff about induction conclusing statements and I am still confused as to which is the 'correct' way.

One of the teachers at my school (very good teacher too if I may add, he certainly isn't that one who'd 'blindly' teach from a textbook) reckons it should be something like "It is true for n=1, so it is true for n=1+1=2. It is true for n=2, so it is true for n=2+1=3 and so on for all positive integral values of n"

I mentioned the stuff I in the 2005 Ext 2 ER that all you need to write is somethihng like "Hence it is true for all n>=1, by induction" and that at university that is the preferred method (I read that somewhere here).

He said that in Ext 2 that may be the case because getting to that step is hard enough, but in Ext 1 he still reckons that I should go with his method. He also said that at university (some 30-40 years ago) he did it that way and got through...

So, any opinions?

 

[jet]

I write:
"Hence, if the statement is true for n=k, then it is true for n=k + 1. Since it is true for n = 1, then it is true for n = 2, 3, 4, 5...... Thus, it is true for n>=1
I just think it makes a cohesive link between n = k, k+ 1 and the use of the initial value. It shows the logic I use as well. I was taught this by my math's teacher, who insisted that his way was the only correct way lol.

 

[Pwnage101]

I was taught:

Therefore we have shown if the statement is true for n=k, it is also true for n=k+1. As it is true for n=1, it is hence true for n=2, hence n=3, n=4, etc. i.e. it is true for all positive integral values of n, by the principle of Mathematical Induction.

 

[kwabon]

i write,
True for n=1, and proven true for n=k+1, therefore proven by induction for all value for n or n is greater than 4 or whatever u like.

 

[buchanan]

jm01, tell your teacher he is wrong.

The following is from the HSC exam committee:

---------------------------

I would like to comment on the induction part of the question.

It has come to my attention that many teachers are training their students to write some form of the following mantra at the end of induction problems.

The statement is true for n=0 and hence is true for n=1. The statement is true for n=1 and hence is true for n=2. The statement is true for n=2 and hence is true for n=3 and so on. Hence the statement is true for all integers n≥0 (by induction).

In many cases the words 'by induction' are omitted.

It needs to be pointed out that

(a) No marks are awarded for this mantra in the marking guidelines for the HSC.

(b) Much time is wasted writing it

(c) Most importantly, the above mantra, especially if the word induction is left out, is at best misleading.

There is a logical (and subtle) difficulty in trying to argue that because the statement is true for any (finite) integer n, it follows that it is true for all non-negative integers n. The axiom of induction is needed to fix this difficulty.

It would be better both mathematically, and for the students themselves, if they ended induction proofs with the simple statement

Hence the statement is true for all n≥0 by induction.

I might add that students who persist in writing this mantra actually LOSE marks in our discrete Mathematics courses at University, so teachers are not doing their students any service, either in the short term (HSC marks) or in the long term. I (and others) have been complaining about this for a long time but without success.

------------------------

And from the Board of Studies on page 9 of http://www.boardofstudies.nsw.edu.au/hsc_exams/hsc2005exams/pdf_doc/maths_ext_2_er_05.pdf :

The setting out of the induction warrants comment. A very large number of candidates who successfully completed the question (and many who attempted it) ended the induction proof with some version of the following:

"The statement is true for n = 0 and hence is true for n = 1. The statement is true for n=1 and hence is true for n=2. The statement is true for n=2 and hence is true for n=3 and so on. Hence the statement is true for all integers n≥0 (by induction)."

In a large number of cases the words "by induction" were omitted. Much time is wasted writing such a lengthy final statement and it would be better if candidates ended induction proofs with a simple statement like:

"Hence the statement is true for all n≥0 by induction."

 

[vds700]

Hi,

I recently read some stuff about induction conclusing statements and I am still confused as to which is the 'correct' way.

One of the teachers at my school (very good teacher too if I may add, he certainly isn't that one who'd 'blindly' teach from a textbook) reckons it should be something like "It is true for n=1, so it is true for n=1+1=2. It is true for n=2, so it is true for n=2+1=3 and so on for all positive integral values of n"

I mentioned the stuff I in the 2005 Ext 2 ER that all you need to write is somethihng like "Hence it is true for all n>=1, by induction" and that at university that is the preferred method (I read that somewhere here).

He said that in Ext 2 that may be the case because getting to that step is hard enough, but in Ext 1 he still reckons that I should go with his method. He also said that at university (some 30-40 years ago) he did it that way and got through...

So, any opinions?

I went to a (3 unit) maths lecture on induction at usyd a few yers back and we were told to write
"Hence the statement is true for n > ... by induction"

 

[buchanan]

And this is from the Chief Examiner of the HSC:

The concluding statements for inductions provided by many candidates show that they incorrectly think that a proof by induction is actually an iterative proof, in which you imagine that the recipe should be repeated as many times as necessary in order to verify the statement for whichever positive integer is of interest. In fact, the Principle of Mathematical Induction is that every set with the property that, for each integer n in the set, n+1 is also in the set and which also contains 1 contains all positive integers. So, having established that the statement is true for 1 and, if true for some integer, is also true for the next integer, the correct conclusion is to simply state that, by induction, the statement is true for all positive integers. At least 75% of teachers disagree with the HSC exam committee. This is because most maths teachers blindly teach from textbooks written by amateurs without thinking about what they are teaching. Not only is this a very boring way to teach, it also will lead to errors such as this one.

 

[LordPc]

The following is from the HSC exam committee:

---------------------------

I might add that students who persist in writing this mantra actually LOSE marks in our discrete Mathematics courses at University, so teachers are not doing their students any service, either in the short term (HSC marks) or in the long term. I (and others) have been complaining about this for a long time but without success.

quoting my Discrete Maths lecturer

"If you are doing an induction proof and dont mention the word 'induction', then something is very wrong"

 

[Pwnage101]

jm01, tell your teacher he is wrong.

The following is from the HSC exam committee:

---------------------------

I would like to comment on the induction part of the question.

It has come to my attention that many teachers are training their students to write some form of the following mantra at the end of induction problems.

The statement is true for n=0 and hence is true for n=1. The statement is true for n=1 and hence is true for n=2. The statement is true for n=2 and hence is true for n=3 and so on. Hence the statement is true for all integers n≥0 (by induction).

In many cases the words 'by induction' are omitted.

It needs to be pointed out that

(a) No marks are awarded for this mantra in the marking guidelines for the HSC.

(b) Much time is wasted writing it

(c) Most importantly, the above mantra, especially if the word induction is left out, is at best misleading.

There is a logical (and subtle) difficulty in trying to argue that because the statement is true for any (finite) integer n, it follows that it is true for all non-negative integers n. The axiom of induction is needed to fix this difficulty.

It would be better both mathematically, and for the students themselves, if they ended induction proofs with the simple statement

Hence the statement is true for all n≥0 by induction.

I might add that students who persist in writing this mantra actually LOSE marks in our discrete Mathematics courses at University, so teachers are not doing their students any service, either in the short term (HSC marks) or in the long term. I (and others) have been complaining about this for a long time but without success.

------------------------

And from the Board of Studies on page 9 of http://www.boardofstudies.nsw.edu.au/hsc_exams/hsc2005exams/pdf_doc/maths_ext_2_er_05.pdf :

The setting out of the induction warrants comment. A very large number of candidates who successfully completed the question (and many who attempted it) ended the induction proof with some version of the following:

"The statement is true for n = 0 and hence is true for n = 1. The statement is true for n=1 and hence is true for n=2. The statement is true for n=2 and hence is true for n=3 and so on. Hence the statement is true for all integers n≥0 (by induction)."

In a large number of cases the words "by induction" were omitted. Much time is wasted writing such a lengthy final statement and it would be better if candidates ended induction proofs with a simple statement like:

"Hence the statement is true for all n≥0 by induction."

true, at uni we've been taught to use a simple conclusion like "Hence the statement is true for all n≥0 by the principle of mathematical induction"

Having said that, i find that at year 12 3Unit level, having the conclusion i wrote above helps students grasp what it is they are doing (proving a result), and the part 'as it wa strue for...' (before you invoke the principle of induction) is more for the students - writing out what is going on in their brain.

It should aslo be taken into account that at year 12 3 unit level, students havent been exposed to the well-oordering principle or the 'strong form' of induction

 

[khorne]

In the cambridge 3U book the statement is:

(It follows from parts A and B) by mathematical induction that the statement is true for all positive integers n.

 

[cutemouse]

jm01, tell your teacher he is wrong.

The following is from the HSC exam committee:

Could you please link me to a direct source so I can show him?

(a) No marks are awarded for this mantra in the marking guidelines for the HSC.

Is this the case with both Ext 1 and Ext 2?

I might add that students who persist in writing this mantra actually LOSE marks in our discrete Mathematics courses at University

I told him that, but he said he didn't lose marks in university when he did it his way. Although it was probably like 40 years ago. Have things changed since then?

 

[MaNiElla]

 

[MC Squidge]

so what do we write after proving true for n=k+1?

 

[MaNiElla]

you write nothing

 

[cutemouse]

so what do we write after proving true for n=k+1?

Something like "If it is true for n=k, so it is true for n=k+1"

 

[khorne]

^^ Include the word induction

 

[Aerath]

LOL, for some reason, the word 'induction' escaped my conclusion, and I still got the mark for it....But yeah, was rather stupid of me.

 

[tywebb]

on the other hand, the oldest example uses the mantra, if true for 1, then true for 2, then true for 3, etc. and didn't use the word induction either.

plato was the first to use it 24 centuries ago (documents attached) and is just as guilty as jm01's teacher for using the dreaded mantra.

one suspects plato is more famous than the hsc exam committee!

 

[gurmies]

LOL, for some reason, the word 'induction' escaped my conclusion, and I still got the mark for it....But yeah, was rather stupid of me.

Haha, I never write induction

 

[the-derivative]

I used to be confused with these statements as well, however at a Catholic Education Office 4U day, I was told by some marker from Reddham House that the statement isn't important.

So now I just write something like 'Hence true py process of mathematical induction.'