A few questions about a math class

Factual Questions
1

Servant, rather then spending the rest of his life leaching off of society, is going back to college. (Contain your enthusiasm.)

Anyway, one of the classes I have to take is College mathematics. The course description says that topics to be covered include, “set theory, logic, the real number system, graphs and functions…”

I know what most of them are but what is set theory? And what is logic as it relates to a math class?

If you could explain to me like I’m a 6 year old that’d be great.*

Thanks.

*Not that I am 6 years old. Far from it. More like 34.

2

Set theory is about sets. Sets are groups of numbers. For example, the set of positive numbers, or the set of integers, or the set containing { 3, 7, 12 and 15 }. Set theory is about how sets are defined, operations you can do on sets, unions and intersections of sets, and so on. You get to make pretty diagrams.

Logic is basically the study of how to determine if things are true or not. You do this by applying various properties to given information to prove something. Example: It’s given that if A is true, then B is true. And it’s given that if B is true, then C is true. You can then prove that if A is true, so is C.

3

Well, set theory in a nutshell is the mathematics of sets of things. Basically the mathematics of categorization of items.

For example, you could talk about the set of all of the socks in your sock drawer (call it Sd), and the set of all of the white socks in your house (call it Sw).

These are distinct mathematical sets. Sets have elements (specific socks). They have a size in terms of the number of elements (not always finite, but chances are you’ll deal with finite sets).

You can perform intersections of the sets, producing another set. Note that Sd and Sw overlap. So Sd intersect Sw is the set of all white socks in your sock drawer, the intersection, or common elements, of both.

You can union the sets, producing another set. Sd union Sw are all of the socks in your drawer plus all of the white socks of the house. Some are counted twice, but you ignore the duplication. You still describe a set of items.

Negating a set produces another set. Negate Sd is the set of all socks not in your drawer. This overlaps with Sd, of course, but also includes all other socks in the universe (assuming we restrict our universe to talking about socks).

There are other operations too, but this isn’t a math course. :slight_smile: Set theory is very powerful because it can be used in more abstract ways to discuss mathematics, to count things, to prove things about collections and categorizations, etc.

For example, you could use it to prove that the set of all integers is infinite (in size), and that the set of all even integers is also infinite, and that oddly enough they are the same size. This is likely to be outside the bounds of your course, but properties like this can be significant.

Proving things being inside or outside of very complex combinations of sets can help you identify its properties, be they physical objects or mathematical entities.

4

Upon preview, people are already explaining it way better than I was going to, so I’ll just add that this kind of logic is used in computer programming, on the off chance that you know anything about that (if/then statements, boolean operations, things like that).

5

The study of logic is the study of basically proofs, but in (typically) symbolic form. Pure logic doesn’t need pesky interpretations of languages to get in the way.

For example:

If Harry failed English, he didn’t go to college
If Harry didn’t get into college, he didn’t make the big bucks.
Harry’s a multimillionaire.

From this we conclude that he got into college and passed English. You could argue eternally about the validity of the actual premises themselves, but that is an English and Economics problem, not a logic problem.

By assigning:

A: Harry fails English
B: Harry doesn’t get into College
C: Harry is a multimillionaire

you can restate the problem as:

1: A -> B (A implies B, or in other words, if A then B)
2: B -> C
3: ~C (not C, the opposite of C)

These are the things we know. From this we deduce:

4: ~B (by way of 2, and 3, and the logic rule of modus tollens)
5: ~A (by way of 1, and 4, and the logic rule of modus tollens)

So we proved ~A, or that Harry passed English, without the ungainly mess of language. Furthermore, the logic is, at a ‘mathematical’ level, correct and infallible.

The beauty of studying logic is that you can reason with it, even informally once you know what can and can’t be done, for the rest of your life, in any scientific or mathematical study. (I took a logic class, and applied it to better prove mathematical problems, and even once being convincing about a history paper).

6

Blast, blew the proof! :smiley:

Let me do it again.

1: A -> B (A implies B, or in other words, if A then B)
2: B -> ~C (B implies ~C, where ~C is not C, the opposite of C)
3: C
These are the things we know. From this we deduce:

4: ~B (by way of 2, and 3, and the logic rule of modus tollens)
5: ~A (by way of 1, and 4, and the logic rule of modus tollens)

7

Hey, I remember being taught set theory when I was six. It was fun.

It wasn’t formalized, of course, but we would put the blocks in groups and then loop string around them (like the diagrams). It was also used as an introduction to multiplication.

8

Yup, those are called Venn Diagrams.

9

panamajack, you probably didn’t learn much actual set theory, you just learned what some deluded educators thought was set theory. It was supposed to make learning math oh-so-simpler, but I’m not sure how well it worked.

10

That’s unfair Chronos, panamajack was six at the time, I doubt that it was the intention that pj come away with an undergraduate understanding of the theory of sets.

These “deluded educators” seem to have hit the nail on the head – set theory is really about classes of object and inclusion rules, asking a six-year old to trace lines around The Set Of Red Things and a different line around The Set Of Big Things and encouraging them to see that those objects contained by both lines constitute The Set Of Big, Red Things is a pretty damned cool thing to do.

I recall amazing afternoons* sat on the floor with the two daughters (three and five years old) of good friends of mine, drawing on a blackboard and arranging objects on the floor to explore ideas of symmetries (or whatever). It might be coincidental but they are both very strong at mathematics now (eight years later).

Payton’s Servant, mathematics is a beautiful, mind-expanding subject, the theory of sets and logic are delightfully intertwined, I am sure that if you give this course the time and effort it deserves that you will find yourself bountifully rewarded.

*really, it sends a tingle down my spine talking about it.

11

Minor hijack - does modern mathematics attempt to define what a set is in any formal manner? When I did my undergrad studies (many moons ago), we learned to not ask awkward questions like, “what is the formal definition of a set?” and just assume that we intrinsically all understand what a set is, then proceed to do useful mathematics without ever having nailed it down.

12

I read a definition that said that a set was a collection where you knew exactly whether a thing was in it or not. So buildings 50’ tall or taller would be a set, where as the set of attractive people wouldn’t be since attractiveness is so subjective. Then the author used that definition to explain why the Axiom of Choice was so dicey.

Regarding the OP, check out: www.dmoz.org/Science/Math/ along with the sub-directories www.dmoz.org/Science/Math/Logic_and_Foundations/ and www.dmoz.org/Science/Math/Logic_and_Foundations/Set_Theory/