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adoni [48]
3 years ago
15

Find (a) the number of subsets and (b) the number of proper subsets of the set.

Mathematics
1 answer:
vaieri [72.5K]3 years ago
3 0

Answer:

(a) Total No. of Subsets = 128

(b) Total No. of Proper Subsets = 127

Step-by-step explanation:

First we need to define the set of days of the week.

Set of Days of Week = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}

It is evident from the set of days of the week, that it contains 7 elements.

(a)

The total no. of subsets of a given set is given by the following formula:

Total No. of Subsets = 2^n

where,

n = no. of elements of the set = 7

Therefore,

Total No. of Subsets = 2^n

Total No. of Subsets = 2^7

<u>Total No. of Subsets = 128</u>

(b)

The total no. of proper subsets of a given set is given by the following formula:

Total No. of Proper Subsets = (2^n) - 1

where,

n = no. of elements of the set = 7

Therefore,

Total No. of Proper Subsets = (2^n) - 1

Total No. of Proper Subsets = (2^7) - 1 = 128 - 1

<u>Total No. of Proper Subsets = 127 </u>

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\huge\boxed{y=-\dfrac{1}{4}x-1\to x+4y=-4}

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\text{Let}\ k:y=m_1x+b_1;\ l:y=m_2x+b_2\\\\l\ ||\ k\iff m_1=m_2\\\\l\ \perp\ k\iff m_1m_2=-1\to m_2=-\dfrac{1}{m_1}\\==========================\\\\\text{We have}\ x+4y=-6.\\\text{Convert to the slope-intercept form:}\\\\x+4y=-6\qquad\text{subtract}\ x\ \text{from both sides}\\\\4y=-x-6\qquad\text{divide both sides by 4}\\\\y=\dfrac{-x}{4}-\dfrac{6}{4}\\\\y=-\dfrac{1}{4}x-\dfrac{3}{2}\to m_1=-\dfrac{1}{4}

\text{Lines are to be parallel. Therefore}\ m_2=-\dfrac{1}{4}.\\\\\text{We initially have the form equation}\ y=-\dfrac{1}{4}x+b.\\\\\text{The line passes through the point}\ (9,\ -3).\\\\\text{Substitute the coordinates of the point to the equation of a line:}\\\\x=9,\ y=-3\\\\-3=-\dfrac{1}{4}(9)+b\\\\-3=-\dfrac{9}{4}+b\qquad\text{add}\ \dfrac{9}{4}\ \text{to both sides}\\\\-\dfrac{12}{4}+\dfrac{9}{4}=b\to b=-\dfrac{3}{4}

\text{Lines are to be parallel. Therefore}\ m_2=-\dfrac{1}{4}.\\\\\text{We initially have the form equation}\ y=-\dfrac{1}{4}x+b.\\\\\text{The line passes through the point}\ (8,\ -3).\\\\\text{Substitute the coordinates of the point to the equation of a line:}

x=8,\ y=-3\\\\-3=-\dfrac{1}{4}(8)+b\\\\-3=-2+b\qquad\text{add 2 to both sides}\\\\-1=b\to b=-1

\text{Therefore the equation is:}\ y=-\dfrac{1}{4}x-1.\\\\\text{Convert to the standard form}\ Ax+By=C:\\\\y=-\dfrac{1}{4}x-1\qquad\text{multiply both sides by 4}\\\\4y=-x-4\qquad\text{add}\ x\ \text{to both sides}\\\\x+4y=-4

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