Set Theory Pre-Calculus?

Question

I started Set Theory today in Pre-Cal and am absolutely lost. My teacher offers no help outside class. Her notes aren't very sufficient and it isn't in our text book.

Determine whether or not each set is the NULL set. What does this mean?

Y = {x: x^2=9, 2x=4}

Y = {x:x≠x}

Z= {x:x+8=8}

If you could give me a basic understanding of set theory it would help tremendously and if you could try to explain these problems.

Also, what is the difference between an elongated C that is underlined and an elongated C with a slash through it? When do you use each when comparing sets? What is a 0 with a slash through it?

Thanks SO much!

Answer 1

Hi Shannan,

1) the null set.

a set is a collection of elements. We can define a set by explicitly enumerating the elements inside. For instance, consider the set of all humans. Then a possible set is {you, me, your teacher}.

Another possible way to define a set is by stating where the elements are drawn from, and stating some properties they need to satisfy.

example: A = {x: x is a positive integer and x<10}. Then we can see that A = {1,2,3,4,5,6,7,8,9}.

Using this representation of a set, consider the following set:

A = {x: x is a number and x>0 and x<-1}. Since there is no number such that, simultaneously, x>0 and x<-1, it follows that A is a set with no elements.

That set is called *the empty set*,, or  *the null set*. It is noted as a 0 with a slash through it.

b) C underlined and with a slash through it.

Consider two sets A and B. Say A = {1,2,5} and B = {1,2,3,4,5}. One can see that all elements of A are elements of B. We then say that A is a *subset* of B. We note that as A C B (where the C is the elongated C with a bar under it), and it is read "A is a subset of B".

We can now see that two sets can be equal. for instance

A = {1,2,3}

B = {x: 0<x<4 and x is an integer}

Then it follows that B = {1,2,3} = A. Here we have that A is a subset of B and B is a subset of A.

Assume now that A C B, but that there exist an element in B that is not in A. Then it follows that B is *not* a subset of A. We then say that A is a *proper subset* of B. We note that A c B, where c is the elongated C with the slash.

Hence, for A={1,2,5} and B = {1,2,3,4,5}, we have A c B.

Note that, for any two sets, if A C B, then A c B. The converse relation is not true.

c) Let's do the examples:

Y = {x: x^2=9, 2x=4}. Note that x^2 = 9 implies that x = 3 or x = -3. Now 2x = 4 implies x = 2. Hence, since a number cannot simultaneously be equal to 3 and 2, or to -3 and 2, it follows that Y is the the null set.

Y = {x: x \neq x}. Since for any number x, x=x is always true, it follows that there is no x such that x \neq x. Hence, again, Y is the null set.

Z = {x: x + 8=8}. Since x + 8 = 8 implies x = 0, it follows that x=0 is an element of Z. Further, x = 0 is the only element of Z, thus we conclude that

Z = {0} (here 0 is the number zero), and thus Z is not the null set.

If you are interested, there are lecture notes for a class in discrete mathematics at Stanford university (the first chapter deals briefly with finite set theory)

http://www.stanford.edu/class/cs103x/

There you can download the lecture notes for all the class

http://www.stanford.edu/class/cs103x/cs1...

Good luck!