Class Review: Sets

On Friday we talked about sets in class.

Definition

A set is just a collection of things, mathematical or otherwise. I could have a set of numbers, symbols, ice cream cones, pandas, whatever.

Say I have a set of: {7, 42, #, h, @, $} Each of the characters in this set is called an element. Since this set also has numbers, symbols, and letters in it, we can also call it a set of mixed types.

Some Common Sets

N = all natural numbers

Z = all integers

Q = all rational numbers

R = all real numbers

These are not the only sets in existence, just the most common.

Another common set we talked about was the null or empty set, which is exactly what you think it is. The set with nothing in it. It’s written as : {   }.

The set {∅} does not mean the same thing. That set is not empty, because it has another set inside of it: the null set.

Set Relationships

Sets can be related to each other as well. For example, all natural numbers are integers. Therefore, the natural numbers are a subset of integers. Similarly, all integers are rational, so integers are a subset of rationals. All rational numbers are real, so rational numbers are a subset of real numbers.

Then we bring complex numbers into the mix. Complex numbers are all real numbers..plus the unreal numbers. This means that we can say that real numbers are a subset of complex numbers, as all real numbers fall into the category of complex. We can write out what complex numbers are made of as:

{a + (i)b: a,b \in \!\, \mathbb{R} \!\,}, so when we’re talking about a real number, b=0.

Union/Intersection

A union of two sets means:

\cup \!\, B = {x: x \in \!\, A or x\in \!\,B}

Which basically means that this new set contains elements of A or B, it doesn’t matter where they come from.

An intersection means:

\cap \!\,B = {x: x \in \!\, A and x\in \!\,B}

Which basically means that this new set only contains elements that are in both A and B.

When you subtract two sets, the new set becomes:

A – B = {x: x \in \!\, A and x∉B}, which basically just a mathy version of realizing that when you subtract two sets, the new set is all of the elements in A, with all elements of B taken away.

You can compare more complicated sets as well. Say we compare 2×2 matrices and all real numbers.

M2x2 \cap \!\, \mathbb{R} \!\, = ∅, because the two sets have nothing in common, so their intersection set contains nothing. The 2×2 matrices may contain real numbers, but the sets are in two different formats, and can therefore not be compared.

Theorems

If B \subseteq \!\, A, then B \cap \!\, A = B.

If B \subseteq \!\, A, then B \cup \!\, A = A.

When you think about it, these two theorems make sense.

If B is a subset of A, that means that all elements in B are elements in A as well, so when only looking at elements in common with both A and B, that is simply all of B.

Since the same holds true for B being a subset of A, and we compare the two sets so that we use all elements part of A and B, this set would just be A, because A is comprised of all of B’s elements plus some unique elements only in A.

Infinity

The cool thing about intersection and union is the fact that you can perform both an infinite amount of times.

For example, if you wanted to compare these sets:

[-1 , 1] \cap \!\, [-1/2 , 1/2] \cap \!\, [-1/4 , 1/4]….(to infinity and beyond)

You would look at what all of these sets that are getting infinitely smaller, and see what elements they share. In these case, this intersection would equal the set {0}, since the sets are getting smaller and smaller.

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One thought on “Class Review: Sets

  1. Pingback: FOM BOOK: 1.3 | FOM is for winners

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