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Hello, this is section 1.
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The topic of the section is
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common sets of numbers and
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set notation.
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So, the goal of this section is
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basically just to orient ourselves
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and build a common foundation
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for learning the language
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of mathematics,
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including set notation,
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set building notation,
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and, specifically, therefore,
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very common sets of numbers
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I want to solve to be aware of
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as we move forward in this unit.
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The first of these sets of numbers
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is written as a capital Z,
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or you could say, boldface Z,
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and that is concise notation
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for the set of all integers.
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Now, integer is just synonymous
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with a whole number, okay?
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So, this capital Z,
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stands for the set of all integers.
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Now, how big is that set?
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Well, that set's an infinite set.
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There's an infinite number of integers.
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In fact, it's a doubly infinite
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array of numbers.
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So I'm gonna write another
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description of that set
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(the set of all integers)
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in what's called set notation.
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So, for set notation,
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you start with a curly brace, okay?
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So a left curly brace,
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and then at the end of this definition
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I have a right curly brace, over here,
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and then between them I'm gonna
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fill in all this concise description
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and all the elements of that set,
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separated by commas.
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Let me show you how this works.
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So we have a dot dot dot.
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I'll talk about what that means,
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and let's just say
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negative three, negative two,
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negative one, zero, one,
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two, three,
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dot dot dot.
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I'm gonna append this
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with a right curly brace.
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Okay.
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So there is set notation
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for the set of all integers.
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What I've done
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is that I'm separating
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the elements
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or, simply, let's call them members
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of the set of all integers
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by commas,
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and on the end here
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I'm sort of bookending this description
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with what's called ellipsis,
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so the dot dot dot just means
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we continue on forever
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on both directions,
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to positive infinity
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to the right and to the left
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with respect to negative infinity.
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That's what I meant by being
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a doubly infinite array.
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Okay.
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Now, I can describe this set.
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I'm gonna draw something called
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a Venn diagram here.
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So let's just say I have this framework.
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I have a universe of sets.
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So, U just stands for universe.
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This is an example of a Venn diagram.
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Within that universe, I have this set,
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to be except, for sure,
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the set of all integers.
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So we just label that.
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It's kind of a bubble,
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and just slap on this Z there,
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just to indicate that set of all integers.
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Now, the question
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I want to ask or pose is:
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Are there numbers
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outside of that set
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that we can imagine?
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And the answer is, of course, yes,
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and there're lots of numbers
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you're probably well familiar with
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that aren't contained
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in the set of all integers.
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I mean, one such example,
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would be like the number 1/2, right?
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1/2 would be floating out here.
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Somewhere.
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I'm just gonna use a sort of dot
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to indicate
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1/2 is out there.
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Certainly it's outside of
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the set of all integers.
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So, on the heels of that,
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let's introduce some little bit
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more notation.
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So, there's set notation.
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Now, we'd like to,
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when we talk about sets,
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be able to coherently describe
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what's called a membership
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in that set.
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So, for instance,
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here's the notation as it works.
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Let's say, the number 4.
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The number 4.
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That's a whole number.
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That's an integer.
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So, I can write this notation.
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So, I would read this as follows.
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This symbol, if your not familiar,
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is epsilon,
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and epsilon here,
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with relation to sets,
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just means "is an element of".
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You could read that as
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4 is an element of the integers,
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or you can say 4 is contained
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within the integers.
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So, yeah, sure enough.
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4 is in the integers.
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That's in the bubble.
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Somewhere in my Venn diagram,
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but 1/2 is not in there.
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So, that's how we denote
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membership in a set.
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"4 is an element of"
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We use the epsilon symbol.
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I could similarly write
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0 is an element of the integers,
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or 8 is an element of the integers,
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and so on.
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How can I show
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something's not an element of it
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using this membership notation?
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Well, for instance,
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we just observed, casually,
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"1/2 is not an element of"
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So, I'm just gonna say
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epsilon with a slash through
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is the typical way to notate this,
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and 1/2 is not an element of,
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remember,
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of the integers.
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In fact, we can see
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almost, really, any fraction
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is not gonna be
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contained in the integers,
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so there's a lot more out there
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to sort of discover.
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That leads us to our next kind of
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natural set of numbers.
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Think of Q as standing for quotient.
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So, capital Q, is again a
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concise notation
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standing for what's usually referred
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as the set of all rational numbers.
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What's a rational number?
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By definition, a rational number is
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just like a ratio.
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It's a fraction.
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The set of all rational numbers is
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the set of all fractions.
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We can write all the
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elements in that set
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in the following fashion.
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I can say Q consists of
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all the elements that look like this
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"a" divided by "b"
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and this vertical line in
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set builder notation
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is read "such that".
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So, the set of all things
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that look like this
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defining membership here
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this criterion
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a and b are integers
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and if we're totally thorough
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we can't divide by zero.
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We can't have zero
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in the denominator.
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So we also have to specify
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that b cannot be zero.
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So there you have it.
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One more comment on the
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integers and the rational numbers
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is they have a nice relationship
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if you think about this.
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The integers are entirely
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contained inside of the rationals.
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Another way to put that is
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the integers are
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a proper subset of the rationals,
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and the way that's written in set notation
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is that you would say
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the integers are contained
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inside of the rationals.
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So this notation
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is what's called a subset
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or you could think of containment.
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So, again, we would just say
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Z is a subset
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is contained within
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the rationals.
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How do I illustrate that
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with a Venn diagram?
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Let's just draw another bigger bubble
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that entirely contains Z.
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Also it's gonna contain this fraction
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1/2 for sure.
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And there, that outer bubble, is my Q
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so Z is entirely contained within it
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That helps to sort of
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connect these things.
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Of course we all know, for instance,
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in terms of numbers
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1 is less than 2.
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We use this kind of relation symbol.
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Well, similarly for sets,
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we can talk about kind of one set
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in the sense of being less
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than another one.
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So, this relation just means
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this set, Z, is contained or smaller
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in the way
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is contained inside of Q.
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Okay.
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So what we're gonna do is
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build up a full sequence
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of the most common sets of numbers
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you'll ever encounter
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in any applied science field
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beginning with Z and Q.
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That's kind of our basis.