Discrete Math: Understanding Sets and Elements

[OrangutanLife]

Hi,

My classes don't start until next week and I am trying to get a head start in my linear algebra, discrete math and calc 3 class!

I am using Discrete Math by Epp 4th edition.

1) I know 4 =/ {4} but why? 4 is a symbol representing the number 4, and {4} is a set with only one element which is the symbol 4. I keep thinking if you evaluate (maybe you can't) the set {4} you get the number 4, so 4 = {4}? What am I missing?

2) How many elements are in the set {1, {1}, {1, {1}}}? The answer is three, but I want to say four and here's why...

2a) Take this question from the book (answer is given). How many elements are in the set {1, {1}}? The answer is two, the symbol 1 and the set with only one element, the symbol 1.

So, for 2) let me count the elements: the symbol 1 (one), the set whose only element is the symbol 1 (two), the set whose elements are the symbol 1(three) and the set whose only elemental is the symbol 1 (four). That's 4 elements, but I have a feeling I am making a mistake here:
{1, {1} }. Which is similar to 2a).

Thanks!

 

[OrangutanLife]

Disregard this... I thought I found the solution to 2), but I didn't.

 

[member 587159]

I didn't take any discrete math classes yet but I'm pretty sure for 1) that {1} is not equal to 1 because the former is a set and the latter is a number, as you stated yourself.

 

[haruspex]

the set whose elements are the symbol 1(three) and the set whose only elemental is the symbol 1 (four)

No, it's the set whose elements are (the symbol 1 and the set whose only elemental is the symbol 1). That is all one element.
Think of the {} as an opaque envelope. You open an envelope to find, inside it, a sheet of paper and another envelope. There were two items in the first envelope. Opening the second envelope may reveal more, but does not change the fact that there were only two things in the first envelope.
Your mistake is in peeking inside the contained envelopes.

 

[OrangutanLife]

No, it's the set whose elements are (the symbol 1 and the set whose only elemental is the symbol 1). That is all one element.
Think of the {} as an opaque envelope. You open an envelope to find, inside it, a sheet of paper and another envelope. There were two items in the first envelope. Opening the second envelope may reveal more, but does not change the fact that there were only two things in the first envelope.
Your mistake is in peeking inside the contained envelopes.

That makes sense. Thank you very much for clearing that up.