ISSS THISSSSS CORRECTTT??????please help

Naddika [18.5K]

All The Subsets

For theset {a,b,c}:

The empty set {} is a subset of {a,b,c}And these are subsets: {a}, {b} and {c}And these are also subsets: {a,b}, {a,c} and {b,c}And {a,b,c} is a subset of {a,b,c}

And when we list all the subsets of S={a,b,c} we get the Power Set of {a,b,c}:

P(S) = { {}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} }

Think of it as all the different ways we can select the items (the order of the items doesn't matter), including selecting none, or all.

Example: The shop has banana, chocolate and lemon ice cream.

What do you order?

Nothing at all: {}Or maybe just banana: {banana}. Or just {chocolate} or just {lemon}Or two together: {banana,chocolate} or {banana,lemon} or {chocolate,lemon}Or all three! {banana, chocolate,lemon}

Question: if the shop also has strawberry flavor what are your options? Solution later.

How Many Subsets

Easy! If the original set has n members, then the Power Set will have 2n members

Example: in the {a,b,c} example above, there are three members (a,b and c).

So, the Power Set should have 23 = 8, which it does!

Notation

The number of members of a set is often written as |S|, so when S has n members we can write:

|P(S)| = 2n

Example: for the set S={1,2,3,4,5} how many members will the power set have?

Well, S has 5 members, so:

|P(S)| = 2n = 25 = 32

You will see in a minute why the number of members is a power of 2

It's Binary!

And here is the most amazing thing. To create the Power Set, write down the sequence of binary numbers (using n digits), and then let "1" mean "put the matching member into this subset".

So "101" is replaced by 1 a, 0 b and 1 c to get us {a,c}

Like this:

abcSubset0000{ }1001{c}2010{b}3011{b,c}4100{a}5101{a,c}6110{a,b}7111{a,b,c}

Well, they are not in a pretty order, but they are all there.

Another ExampleLet's eat! We have four flavors of ice cream: banana, chocolate, lemon, and strawberry. How many different ways can we have them?Let's use letters for the flavors: {b, c, l, s}. Example selections include:{} (nothing, you are on a diet){b, c, l, s} (every flavor){b, c} (banana and chocolate are good together)etcLet's make the table using "binary": bclsSubset00000{}10001{s}20010{l}30011{l,s}...... etc ..... etc ...121100{b,c}131101{b,c,s}141110{b,c,l}151111{b,c,l,s}

And the result is (more neatly arranged):

P = { {}, {b}, {c}, {l}, {s}, {b,c}, {b,l}, {b,s}, {c,l}, {c,s}, {l,s}, {b,c,l}, {b,c,s},
{b,l,s}, {c,l,s}, {b,c,l,s} }

SymmetryIn the table above, did you notice that the first subset is empty and the last has every member?But did you also notice that the second subset has "s", and the second last subset has everything except "s"? In fact when we mirror that table about the middle we see there is a kind of symmetry.This is because the binary numbers (that we used to help us get all those combinations) have a beautiful and elegant pattern.A Prime Example

The Power Set can be useful in unexpected areas.

I wanted to find all factors (not just the prime factors, but all factors) of a number.

I could test all possible numbers: I could check 2, 3, 4, 5, 6, 7, etc...

That took a long time for large numbers.

But could I try to combine the prime factors?

Let me see, the prime factors of 510 are 2×3×5×17 (using prime factor tool).

So, all the factors of 510 are:

2, 3, 5 and 17,2×3, 2×5 and 2×17 as well, and2×3×5 and 2×3×17 and ..... aha! Just like ice cream I needed a Power Set!

And this is what I got:

2,3,5,17SubsetFactors of 51000000{ }110001{17}1720010{5}530011{5,17}5 × 17 = 8540100{3}350101{3,17}3 × 17 = 51 ... etc ...... etc ...... etc ...151111{2,3,5,17}2 × 3 × 5 × 17 = 510

And the result? The factors of 510 are 1, 2, 3, 5, 6, 10, 15, 17, 30, 34, 51, 85, 102, 170, 255 and 510 (and −1, −2, −3, etc as well). See the All Factors Tool.

Automated

I couldn't resist making Power Sets available to you in an automated way.

So, when you need a power set, try Power Set Maker.

6 0

3 years ago

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Can you help me create a polynomial function matching the criteria on the left please and thanks

Olenka [21]

Answer:

$f(x) = \dfrac{1}{8}(x + 4)^2(x + 1)(x - 3)$

Step-by-step explanation:

From inspection of the graph, we can see that the curve intercepts the x-axis at (-4, 0), (-1, 0) and (3, 0)

Therefore,

x = -4 ⇒ x + 4 = 0

x = -1 ⇒ x + 1 = 0

x = 3 ⇒ x - 3 = 0

Because (-4, 0) touches the x-axis, then (x + 4)² will be a factor

So (x + 4)², (x + 1) and (x - 3) are all factors of the polynomial

$\implies f(x) = \dfrac{1}{n}(x + 4)^2(x + 1)(x - 3)$

If we multiply the constants, this will give us the y-intercept:

⇒ 4² x 1 x -3 = -48

From inspection of the graph, the y-intercept is -6

So to get from -48 to -6 we need to multiply -48 by 1/8

Therefore, n = 1/8

$\implies f(x) = \dfrac{1}{8}(x + 4)^2(x + 1)(x - 3)$

7 0

2 years ago

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(20 points) please give me motivation to study and study tips or test taking tips for finals. it would be greatly appreciated.

Daniel [21]

Answer: Think about graduating. Think about never having to take the courses again. You're almost at the finish line! It'll be worth it. You've worked hard all year for this. You can do it!

Study tips: I would recommend Quizlet! They have a section that generates study games. It's a lot more fun than normal studying. It's also a good idea to make a goal for yourself. Try to make a challenge of achieving a certain score! By the time you accomplish said score, you'll find that you've learned a lot. Another tip is to make sure you take breaks. If you work too long without giving yourself a break, it will become harder to focus and your brain will become tired. Just don't get too distracted! set yourself an alarm during break times to help you stay on task. If you become frustrated with a certain subject or task, take a break from that task. Use this time as an opportunity to work on another subject. You can begin working on the first subject again once you feel refreshed. A lot of this may sound redundant, but hopefully it will help at least a little bit. Good luck!

6 0

3 years ago

Help plz will give brainliest

grin007 [14]

Answer:t the farthest city south is bellville and the volcano name is sue

Step-by-step explanation:

7 0

3 years ago

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So if a video that is 15:34 how long is it when 2 times speed

rusak2 [61]

Answer:

7:47

Step-by-step explanation:

Here are two ways of solving this:

The video will be half the time of 15:34, so you need to divide the time 15:34 by 2.

Method 1)

Use the tip. The video is 15 minutes and 34 seconds long.

Convert minutes to seconds: 15 * 60 = 900

Add 34 seconds to the 15 minutes: 900 seconds + 34 seconds = 934 seconds

The video is 934 seconds long.

Divide the total seconds by 2: 934/2 = 467

467 seconds = 420 seconds + 47 seconds = 7 minutes and 47 seconds

Answer: 7:47

Method 2)

Divide the minutes by 2 and divide the seconds by 2.

15 minutes and 34 seconds.

Divide minutes by 2: 15/2 = 7.5

Divide seconds by 2: 34/2 = 17

Now we need to add 7.5 minutes and 34 seconds.

0.5 minutes is half a minute, so it's 30 seconds.

7.5 minutes + 17 seconds =

= 7 minutes + 0.5 minute + 17 seconds

= 7 minutes + 30 seconds + 17 seconds

= 7 minutes + 47 seconds

= 7:47

Answer: 7:47

6 0

3 years ago