Math Is Fun Forum

Au101

Member

Registered: 2010-12-01

Posts: 353

Set Builder Notation

Hi Guys,

Does anyone have any tutorials on how to use set-builder notation. I know the basics, but there are still a few things which I'm not 100% clear about.

For example, for one question, I needed to show that 'for all values of a, b and c, the matrix A is singular' and this how I tried to translate that into set-builder notation, but I don't know if I've done it correctly:

Similarly, for another question, I needed to show that 'for all real values of x, the matrix A is non-singular' and this was my attempt at a translation:

Bob

Administrator

Registered: 2010-06-20

Posts: 10,583

Re: Set Builder Notation

hi Au101

A matrix is singular if its determinant is zero.  I'm not sure what you are meaning by a, b and c here.

For your second part, again what is x ?

If a matrix is non-singular it will have an inverse.

Try constructing an inverse.  There must be some constraint on x; which presumably => an inverse exists.

If you are still unsure, you'll need to post the whole question.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Au101

Member

Registered: 2010-12-01

Posts: 353

Re: Set Builder Notation

Hi Bob,

Thanks, once again, for replying when I'm in need of help. I don't think I made my question clear, however - for which I do appologise. I was able to do those questions with a reasonable degree of ease, but, since they are 'show that' questions, I wanted to finish my answer with a mathematical conclusion. So, let's take my first question as an example, my approach was to find the determinant of the given matrix A, which, when evaluated came out to be abc-abc. What I then wanted to state is:

Therefore, for all values of a, b and c, the matrix A is singular.

Which, of course, follows from the result that the determinant is equal to abc-abc and is what the question had asked me to show.

However, I wished to state it in set builder notation.

My actual question, then, was about set-builder notation, I have a good idea of the workings of it, but expressions such as 'a, b and c are all elements of the set of real numbers' and quantifying over natural language sentences, as opposed to mathematical expressions are examples of things which I do not fully understand. So I was asking if anybody has any good articles or books on set builder notation (or even set theory in general) which I could have a look at, or has the time to give a quick tutorial in set-builder notation, such that I can fix problems similar to the ones above. These were merely examples of the kind of thing which I was struggling to express in set-builder notation.

Thanks

Last edited by Au101 (2010-12-27 12:52:57)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,583

Re: Set Builder Notation

hi Au101

I've never actually called it 'set builder notation, but I know what you mean.

Have a look at:

http://en.wikipedia.org/wiki/Set-builder_notation

For me, it's just a sentence but written in the language of maths.

So if you want to say 'S is the set of matrices where the determinant is zero,  for all values of a, b and c in the real numbers'

how about

when

               (You will have to substitute the correct matrix here.)

Then

The format is

<name of set> = { <restrictions on variables> such that <formula defining set> }

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Au101

Member

Registered: 2010-12-01

Posts: 353

Re: Set Builder Notation

Thanks Bob!