Set Theory

Beginnings of Set Theory…

A set is a collection of well distinguished objects of our conception or imagination. (Cantor)

We make the argument here at Mockingbird Academy, that we can’t create a set without having a reason for it.

A set is a collection of elements, and sometimes the word member is used instead of the word element.

We use capital Latin alphabet letters for the symbols for sets.

A={red, yellow green}

B={a,e,i,o,u}

C={heads, tails}

In the above examples we define sets by simply showing all the elements in them. Often it is more convenient to define a sey by describing what is in it rather than by listing all its elements.

{“all the counting numbers less than a thousand”}

The above example is just to give you the idea. We want you to learn the notation that is used for this activity.

A={x \in N | 1 < x < 6}[latex] </p> <ul class="wp-block-list"><li>The N signifies natural numbers</li><li>[latex]x \in N reads as "x is an element of N

  • The restriction of being greater than 1 and less than 6 is the last information we need, to know what is in the set

    • {2,3,4,5}

    E={x \in N | x = 2y, y \in N}

    The above set, E, is the set of positive numbers.

    A set that contains all the elements, subject to our arbitrary decree, is called the Universal Set.

    We use the Universal Set to decree what elements are in the universe.

    Let F={1,2,3,4,5}

    Let G={1,3,5}

    G is a subset of F. We can share that with notation as follows:

    G \subset F

    Appendix A

    You may have noticed that when we talk about things that we want to have very technical definitions, we have to come up with vague words that don't have any technical meaning to try to teach the technical idea.

    For example, in order to give you the idea of a 'set' we use the word "collection" and when we wanted to give you the idea for an 'element' of the set we use the word "thing".

    We said "the things in a collection" to take you to the idea of "the elements of a set".

    Some of your associations are rather desperate, but we hope that they work: "a force is a push."

    After you've done quite a bit of work with physics, you would argue back that a force is only one example a force. However, we had to start somewhere.