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My name is Ann [436]
3 years ago
9

Let abe a finite set and let a∈a. prove that the number of subsets of a that contain a1 equals the number of subsets of a that d

o not contain a1.
Mathematics
1 answer:
vladimir1956 [14]3 years ago
5 0
Here's a combinatorial proof. Suppose A has n elements.

For a subset to contain a, it must consist of at least one element. So if any given subset has k elements, where 1\le k\le n, then a is not one of the other k-1 elements. This means the number of subsets containing a is

\displaystyle\sum_{k=1}^n\binom11\binom{n-1}{k-1}

Put another way, we are choosing elements from A to form a subset of k elements. We want a to be in each subset, so we have n-1 other elements of A from which to choose. Then we sum over all the possible sizes of the desired subset.

On the other hand, if we want to build subsets not containing a, then we have n-1 total elements to choose from, and we can make subsets of size ranging from 0 to n-1, so the number of subsets not containing a is

\displaystyle\sum_{k=0}^{n-1}\binom10\binom{n-1}k

We have \dbinom10=\dbinom11=1, and in the second sum we can shift the index up by 1 to get

\displaystyle\sum_{k=1}^{n-1+1}\binom10\binom{n-1}{k-1}

which is the same as the first count.
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Simplify: −√16x^2 , when x<0
Savatey [412]

Answer:

-4|x|

Step-by-step explanation:

To simplify :

-\sqrt{16x^2} , when x

In order to simplify the given expression we will take the square root of the term inside the root.

The term in the square root are perfect squares (square of an integer), so, we can write them in square of the numbers.

⇒ -\sqrt{(\pm4x)^2}   [As (\pm4x)^2=16x^2]

On taking square root the square gets removed. So, we have,

⇒ -(\pm4x)

⇒ -4|x|

Since x, so the value would be negative.

So, simplified answer is

-4|x|  (Answer)

3 0
3 years ago
Please help with this geometry question
Andrew [12]

It has been proven that of all line segments drawn from a given point not on it, the perpendicular line segment is the shortest.

<h3>How to prove a Line Segment?</h3>

We know that in a triangle if one angle is 90 degrees, then the other angles have to be acute.

Let us take a line l and from point P as shown in the attached file, that is, not on line l, draw two line segments PN and PM. Let PN be perpendicular to line l and PM is drawn at some other angle.

In ΔPNM, ∠N = 90°

∠P + ∠N + ∠M = 180° (Angle sum property of a triangle)

∠P + ∠M = 90°

Clearly, ∠M is an acute angle.

Thus; ∠M < ∠N

PN < PM (The side opposite to the smaller angle is smaller)

Similarly, by drawing different line segments from P to l, it can be proved that PN is smaller in comparison to all of them. Therefore, it can be observed that of all line segments drawn from a given point not on it, the perpendicular line segment is the shortest.

Read more about Line segment at; brainly.com/question/2437195

#SPJ1

7 0
1 year ago
Please help I'll give brainliest
Liula [17]

the answer is going to be 6.0

Step-by-step explanation:

1.5 + 1.5 = 3

3 + 3 = 6

6 0
3 years ago
Frank has a 2-digit number on his baseball uniform. The number is a multiple of 10 and has 3 for one its factors. What three num
Effectus [21]
30, 60, 90

a. What possible jersey numbers does Frank wear.
b. its a multiple 10 and has 3 for one of its factors and its a 2-digit number,
c. By eliminating all the incorrect answers?
d. Number on, 10, 3, and frank could have 30, 60, or 90.
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3 years ago
If y varies inversely as x and y = 4 when x = 12, find the constant of variation
Artyom0805 [142]
X is not equal to the answer so y is incorrect
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3 years ago
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