Question about sets

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johnnytruant

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Hello,

New to the forum. Hope this is the right place for this question.

I'm reading How to Think Like a Mathematician by Kevin Houston.

I'm reading about sets, and he points out that {5} and 5 are not the same things. Can anyone explain what it means to have a set that contains only one element, and why you can't simply treat it as the one element?

Thanks much
 
A set contains elements, also called objects. The set can contain any number of elements from 0 to infinity, ie 0, 1, 2, 3, ...
We list the elements in the set between set brackets, {}.

{ } is the set that has no elements.
{2} is the set that contains one element, namely the symbol for two.
[1,2,4} is the set that contains three elements, namely 1, 2 and 5.

{5) is the set that contains only the element 5.
5 is NOT a set!

You can define a set without using set brackets by describing it in words. Consider the set that contains 1, *, 2/3 and 7/4. This is the same as {1, *, 2/3, 7/4}
 
If I gave you a bag containing a tile with the number 5 written on it, that is different from giving you the tile without the bag. In particular, you can do things with the former that you can't do with the latter (such as count the things inside it).
 
johnnytruant said:
I'm reading How to Think Like a Mathematician by Kevin Houston.
I'm reading about sets, and he points out that {5} and 5 are not the same things. Can anyone explain what it means to have a set that contains only one element, and why you can't simply treat it as the one element?
Click to expand...
Full disclosure, I do not know this book. However HERE is the pedigree of its author.
What is a number? Well that requires an axiom: There exists a set that has zero elements.
In reply #3, that is the empty bag. The a bag containing that empty bag is the number one.
The bag containing that bag is the number two. etc
I encourage you to get the book NAIVE SET THEORY by Paul Halmos. You can find a good cheap used edition.
It is known a the working mathematician's handbook.
 
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