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Nana76 [90]
3 years ago
7

What are all the subsets of {5,9,13}

Mathematics
2 answers:
UNO [17]3 years ago
7 0
(5),(9),(13),(5,9),(9,13),(5,13),(5,9,13) and empty set
Naddika [18.5K]3 years ago
6 0
All The Subsets

For theset {a,b,c}:

<span>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}</span>

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?

<span>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}</span>

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 <span>2n</span> 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:

<span><span> abcSubset</span><span>0000{ }</span><span>1001{c}</span><span>2010{b}</span><span>3011{b,c}</span><span>4100{a}</span><span>5101{a,c}</span><span>6110{a,b}</span><span>7111{a,b,c}</span></span>

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

Another Example<span>Let'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:<span>{} (nothing, you are on a diet){b, c, l, s} (every flavor){b, c} (banana and chocolate are good together)etc</span></span>Let's make the table using "binary":<span><span> bclsSubset</span><span>00000{}</span><span>10001{s}</span><span>20010{l}</span><span>30011{l,s}</span><span>...... etc ..... etc ...</span><span>121100{b,c}</span><span>131101{b,c,s}</span><span>141110{b,c,l}</span><span>151111{b,c,l,s}</span></span>

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} }


<span><span>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"?</span><span>  </span><span>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.</span></span>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:

<span>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!</span>

And this is what I got:

<span><span> 2,3,5,17SubsetFactors of 510</span><span>00000{ }1</span><span>10001{17}17</span><span>20010{5}5</span><span>30011{5,17}5 × 17 = 85</span><span>40100{3}3</span><span>50101{3,17}3 × 17 = 51</span><span> ... etc ...... etc ...... etc ...</span><span>151111{2,3,5,17}2 × 3 × 5 × 17 = 510</span></span>


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.

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Answer:

1. 3 7/12 (decimal: 3.583333)

2. -5 3/28

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4. −1  9 /14

7 0
3 years ago
Find three consecutive integers whose sum is 96.​
levacccp [35]

Answer:

31,32,33

Step-by-step explanation:

31+32+33 = 96

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3 years ago
How to reflect over y axis in an equation?
iren [92.7K]
<span>1)  y = -f(x) (This is the reflection about the x-axis of the graph y = f(x).)  That is for every point (x, y) there is a point (x, -y).
</span><span>2)  y = |f(x)| means that the entire graph will be above the x-axis.  Why?  (The absolute value is always positive, that's why!!)<span>  To graph the absolute value graph, graph the function y = f(x).  Anything above the x-axis, stays above it, anything below the x-axis is reflected above the x-axis and anything on the x-axis, stays on the x-axis.
</span></span><span>3)  y = f(-x)  (This is reflection about the y-axis of the graph y = f(x))  For every point on the right of the y-axis, there is a point equidistant to the left of the y-axis.  That is for every point (x, y), there is a point (-x, y).  
</span><span>4)  Reflections about the line y = x is accomplished by interchanging the x and the y-values.  Thus for y = f(x) the reflection about the line y = x is accomplished by x = f(y).  Thus the reflection about the line y = x for y = x2 is the equation x = y2. </span>
7 0
3 years ago
List and explain the elements that make up the structure of a fugue, then list and explain at least four techniques Bach utilize
mina [271]

Answer:

The fugue begins with an exposure. After the exhibition the alternative composer between episodes and presentations of the subject. In each fugue this scheme is developed in a varied way according to the invention and the needs of the composer.

Step-by-step explanation:

The elements that make up a musical fugue are:

The exhibition: It begins with one of the voices presenting the subject (name given to the theme on which a fugue is based). A second voice continues with the answer. The other voices continue alternately between subjects and responses. The exhibition concludes once all the voices have presented the subject or the answer.

While the second voice presents the answer the first voice makes a counterpoint to this answer. Similarly, when the third voice enters, the first two accompany it with other counterpoints. This counterpoint can be called counterpoint or free counterpoint. It is called a countersubject when it appears next to the answer. When the counterpoint is different in each presentation it is known as a free counterpoint.

The final section of the fugue begins when the subject returns to the initial hue of the fugue and from here to the end of the work. This ending is usually formed in several measures added to the main structure. This conclusion is the coda.

In the first place, he must remember that Bach was a self-taught, he was trained alone and specifically by the study of music itself, by the observation and analysis of other people's compositions, very often of poor significance in front of his own and that he copied, however, of his handwriting. The pedagogical activity of Bach and what he taught: "We must make the fugue of the subject", this means that each subject requires his fugue. Bach added another teaching, "voices must not enter without having something to say, or call before they have said everything they have to say," which means that they have voices should not speak for speaking and that they have no obligation to be always present.

In conclusion, Bach did not give any norm about the fugue, not least in terms of the genre of procedures for cases where this kind of compositions could be applied, leaving subjects free to perform this type of music.

6 0
3 years ago
a red car is traveling 15 miles per hour faster than a blue car. if the red car travels 180 miles in the time needed for the blu
vlabodo [156]
Their distance differs by 15 miles for each hour of travel time. Thus, the travel time of interest is
.. (180 -144) miles/(15 miles/hour) = 2.4 hour

The red car's speed is 180 mi/(2.4 h) = 75 mi/h.
The blue car's speed is 144 mi/(2.4 h) = 60 mi/h.
5 0
3 years ago
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