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Robert McKeown: Any number multiplied by zero is
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just equal to zero. So when we're given a
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question like this, it says find all the real
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zero zeros of a polynomial function. So it's
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saying what find the values of x, such that g of
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x is equal to zero? Well, there's a few things.
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So all I have to do is show that one of these
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factors is equal to zero. And I've shown that g
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of x itself is equal to zero. So here with this
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one, if x is equal to zero, then g of x will be
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equal to zero, no matter what the the other
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values are. So that was one What about the
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second factor? Well, I can see that if x is
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equal to four, then x squared is equal to 16.
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But wait a second, it's plus 16. Is there any
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way to get to square a number and have a
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negative value? know there's no, there's no
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possible real number that I can plug into x
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square plus 16 and come up with a zero value? So
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there's no way no way to make x squared plus 16
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equal to zero? No way. So there's no solution
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for that factor anyway. What about three? This
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one's a little more promising because we have a
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negative sign there. If x is equal to one, then
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g of x, or I should say g of one is equal to one
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squared minus one, which is just equal to zero.
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So if x is equal to one, g of x will be equal to
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zero. Is there any other value? Well, what if x
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was equal to negative one?
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Negative one square
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minus one
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is equal to zero as well. So there are three
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answers to the question if x is equal to zero,
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if x is equal to negative one, and if x is equal
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to one, then g of x is going to be equal to
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zero. Now let's go to ALEKS. And see if we did
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it correctly, I've gone ahead and added the
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values into ALEKS. So we've got negative one,
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zero and one, and I'm going to click on the
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check button. And we got the correct answer.
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We've been given a function f of x, which we
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know is the negative of the absolute value of x.
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And that represents the graph in a blue color
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with the origin coordinates 00. Another function
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has been drawn an H function whose highest point
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is at coordinates negative four, three. And the
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question is asking us to write down the
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expression for the function h of x. So let me
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write out F of X, which is equal to negative the
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absolute value of x. And there's an implication
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here that it's plus zero. Why do I know that
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because if x is equal to zero,
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we get the coordinates 00. Which takes us right
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to here and we know just looking at the diagram,
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the graph we know that that's the maximum. Now
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we want an equation for h of x, and it's gonna
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look something like this. Well, I'll read it
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very generally.
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Or maybe Alright as negative the absolute value
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of x plus a some unknown variable A plus B. And
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so we want to essentially solve for A and B.
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Remember that a is going to move the diagram
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left and right, B is going to move, or I
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shouldn't say the diagram, it's going to move
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the function left and right, and B is going to
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move the function up and down. Now I know that
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the H function has a max at the coordinates
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negative four, three,
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to move the max up or make it to move the max up
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three units. set B equal to well, its original
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value zero, plus three. So set B equal to three
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means I'm gonna have h of x is equal to negative
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x plus a plus three. Let's call this one. And
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this is kind of our step one here. And step two.
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Notice h of x is maximized when x is equal to
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negative four,
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so let's set x equal to negative four.
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There's a couple different ways I could show you
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how to do this, let's recognize that this
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function is equal to our Y axis value. If I do
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that, I can rewrite this whole thing as y is
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equal to what it's equal to. Three, which is
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also equal to negative the absolute value of
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negative four plus a plus three. And if I
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subtract three off both sides of the equation,
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I'm going to get zero is equal to negative
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negative four plus a. What value of a makes this
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true? Well, a has to be equal to four. And now
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we can add everything together. And we see that
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h of x is going to be equal to negative f of x
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plus four, plus three, or h of x is equal to
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negative
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x plus four, plus three. Looking at ALEKS, I'm
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going to press negative and then I'm going to
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touch this little button over here, which is
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going to create an absolute value sign for me.
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I've got x minus four. Excuse me, that's x plus
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four, plus three. And let's see if we've got the
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right expression for h of x. I'll click the
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check button. And we got the right answer. And
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this is a great question, because it shows you a
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more general way of inputting values into a
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function. You can take a function and input it
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into another function, as this example
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demonstrates. This comes up a lot in calculus.
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This comes up a lot in economic applications. A
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lot of students are not familiar with it. It's
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not hard. And so but students since they're not
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familiar with it, they often panic and don't
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understand what's happening. So let's look at
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this question. The question says, here's a
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function g of x, find g of five z. So everywhere
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There's an X, I'm going to plug in five z.
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So we've got two times five z squared minus one.
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Now I'm going to simplify or expand, I guess I'm
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going to expand this expression. And I've got
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five squared. C squares, notice that the
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exponent comes into both the factors, both the
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five and the Z are going to be squared, and the
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minus one stays outside of everything. When I
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work through oops, getting ahead of myself. 50 c
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square forgetting the minus one, I'm going a
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little too fast. But there is the answer. So if
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we want to evaluate g of x, at five z, we can do
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that. Plug it in everywhere there's an X, you're
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going to plug in a five z.
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Now I've entered our answer on ALEKS. And I'm
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going to click the check button. And we got the
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correct answer. We're going to take a look at
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cubic functions. It's important to be familiar
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with cubic functions and the shape that they
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take on. This question is similar to one we did
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a little bit earlier where we had to evaluate a
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number of points using the function and then
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graphing it. And so I'm not going to take too
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much time redoing that again. But I want to make
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sure that you understand and you know how to use
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the ALEKS tool and how to graph a function like
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this. Now, I've gone ahead and solve for it.
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Here are the solutions. Notice that we've got
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fractions for in this case is represented as a
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decimal, but it's a fraction. Or it could be
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represented as a fraction. And the problem with
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this is, it's going to be awfully hard for you
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to find the point negative 1.75 on the ALEKS
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diagram. So let me show you again, the tool that
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we're going to use to solve a question like this
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on ALEKS. Here's the question on ALEKS. And I'm
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going to use the plot anywhere function. So I'm
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going to click on this highlighted little image
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of a diagram. And it's asking me just to show
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you know, it says enter the x&y coordinates of
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the point. And now I've got the coordinates over
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here, to the right of the screen. So that's
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going to make it easy for me to input them in.
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So we've got negative two, negative 14. And then
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I click plot point, negative one, negative 1.75.
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Oops, there's a mistake, make sure I move over
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1.75. And I'm going to plot the point
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zero and zero. That's an easy one, plot the
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point, one, and 1.75. plot the point, and two,
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and 14. And I'm going to plot the point. So it
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looks like I did it properly. Hopefully, I
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didn't make an input error. I'm going to close
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this box here. And then I'm going to click on
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these little axes with the curvy line through
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them. And ALEKS should know that how I want the
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cubic function to be drawn. Now I'll move back
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up so you can see the question. And I'm going to
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click on the check box. And happily, I did it
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successfully.
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Let's take a look at set theory. Set Theory
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theory is really, really quite fun. And it's
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really, really important for understanding how
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to read mathematics, how to communicate
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mathematics. I'll show you this example. It's
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really, really easy. But the key is
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understanding the language of mathematics. If
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you understand the language of mathematics,
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you'll be able to read your textbook. You'll be
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able to read academic papers and really
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understand what those authors are trying to tell
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you. And you'll be able to follow those
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instructions. And you're going to be, that's a
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very employable skill. That's a really great
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skill to have. So let's take a look at a simple
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example that I put together. The question says,
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find the union of dark haired people and tall
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people, then find the intersection of dark hair
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and tall people. And we're given two sets. We've
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got our sets. We've got our dark haired set and
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our tall set. Now, when it says find the union,
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what that means in mathematical language is find
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the people who are either tall, or have dark
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hair. So in this example, well, Ahmed has dark
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hair, so he's in that union. zoo, is in that
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union. They have dark hair. Mo has dark hair.
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Frank is tall. So even though he doesn't have
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dark hair, that's okay. He's in the union of
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dark haired and tall people. And zoo is also
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tall. But we've already included them in the
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Union, and we don't need to record them twice.
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There's nothing wrong with recording them twice.
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But generally, it's better if we just record
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them once, because we don't want to count them
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twice. If we were adding up all the numbers of
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people in this set. And Coco, Coco is tall. She
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doesn't have dark hair. But that's okay. She's
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in the Union. She's still in that union of dark
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hair and tall people. Now, what about the
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intersection? Well, the intersection of dark
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haired and tall people are those who are both
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tall and have dark hair. And I guess I could say
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about the union is that we'd say that this thing
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up here is equal to
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dark you tall at the union of dark hair, and
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tall people is equal to this set with five
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names. Now find the intersection of people who
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are both tall and have dark hair. So if I want
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dark, an upside down you that's a little bit
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bigger than a you this, this symbol. The dark,
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the intersection of dark and tall people is
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going to be equal to well, Ahmed is only has
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dark hair. And he's not tall, so he's not tall.
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Xu has both dark hair and is tall. Moe has dark
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hair, but he's not tall. Frank is tall, but he
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does not have dark hair. And Coco is tall, but
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she does not have dark hair. So the intersection
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of these spaces is just zoo.
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Now here's the question that we see on ALEKS. So
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we're given two sets. The sets contain unknown
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variables represented by the letters. And we're
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asked to find the union of G and M. and then
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find the intersection of G and M. So applying
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what we learned in the previous example, the
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union of G and m is going to be equal to well,
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any of these variables that occur in either the
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set G or the set M. So we're gonna have D, E, F,
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G, H, and J. Now, I tried to keep it in
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alphabetical order, because that's the proper
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way to represent it. It's probably not
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necessary, but it's the it's the appropriate
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it's the best way to represent it. And so the
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union of G and M has six different variables in
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it. When we look at the intersection well,
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neither of these variables to pay And both the
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set G and the set m. so we can say that the
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intersection of these two sets is empty, or it
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is the null set. It's the empty set. So it's
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empty. Or it sounds a little bit cooler in
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English to say it's the null set. But you can
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certainly call it the empty set that's
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completely legitimate. Here's our question on
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ALEKS. So the union of these two sets, I'm going
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to press this little button here to get the
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swirly brackets represented as a set.
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And so the union it has six variables in it. The
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intersection is the empty set. And let's see if
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I did it correctly. And I didn't that's okay. I
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just made a mistake with the language of ALEKS.
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And that language is I shouldn't have squirrely
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brackets around by empty set. So I've got the
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empty set here without squirrely brackets and
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now it tells me that I got the correct answer.
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So remember that when you're answering questions
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on ALEKS, it wants the null set with no
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squirrely brackets around it.