All categories

2 years ago

This list shows the number of years that 99 U.S. Supreme Court Justices served.

Mathematics

1 answer:

Darina [25.2K]2 years ago

6 0

Answer:

Step-by-step explanation:

You might be interested in

Which of the following functions are homomorphisms?

Vikentia [17]

Part A:

Given $f:Z \rightarrow Z,$ defined by $f(x)=-x$

$f(x+y)=-(x+y)=-x-y \\ \\ f(x)+f(y)=-x+(-y)=-x-y$

but

$f(xy)=-xy \\ \\ f(x)\cdot f(y)=-x\cdot-y=xy$

Since, f(xy) ≠ f(x)f(y)

Therefore, the function is not a homomorphism.

Part B:

Given $f:Z_2 \rightarrow Z_2,$ defined by $f(x)=-x$

Note that in $Z_2$ , -1 = 1 and f(0) = 0 and f(1) = -1 = 1, so we can also use the formular $f(x)=x$

$f(x+y)=x+y \\ \\ f(x)+f(y)=x+y$

and

$f(xy)=xy \\ \\ f(x)\cdot f(y)=xy$

Therefore, the function is a homomorphism.

Part C:

Given $g:Q\rightarrow Q$ , defined by $g(x)= \frac{1}{x^2+1}$

$g(x+y)= \frac{1}{(x+y)^2+1} = \frac{1}{x^2+2xy+y^2+1} \\ \\ g(x)+g(y)= \frac{1}{x^2+1} + \frac{1}{y^2+1} = \frac{y^2+1+x^2+1}{(x^2+1)(y^2+1)} = \frac{x^2+y^2+2}{x^2y^2+x^2+y^2+1}$

Since, f(x+y) ≠ f(x) + f(y), therefore, the function is not a homomorphism.

Part D:

Given $h:R\rightarrow M(R)$ , defined by $h(a)= \left(\begin{array}{cc}-a&0\\a&0\end{array}\right)$

$h(a+b)= \left(\begin{array}{cc}-(a+b)&0\\a+b&0\end{array}\right)= \left(\begin{array}{cc}-a-b&0\\a+b&0\end{array}\right) \\ \\ h(a)+h(b)= \left(\begin{array}{cc}-a&0\\a&0\end{array}\right)+ \left(\begin{array}{cc}-b&0\\b&0\end{array}\right)=\left(\begin{array}{cc}-a-b&0\\a+b&0\end{array}\right)$

but

$h(ab)= \left(\begin{array}{cc}-ab&0\\ab&0\end{array}\right) \\ \\ h(a)\cdot h(b)= \left(\begin{array}{cc}-a&0\\a&0\end{array}\right)\cdot \left(\begin{array}{cc}-b&0\\b&0\end{array}\right)= \left(\begin{array}{cc}ab&0\\-ab&0\end{array}\right)$

Since, h(ab) ≠ h(a)h(b), therefore, the funtion is not a homomorphism.

Part E:

Given $f:Z_{12}\rightarrow Z_4$ , defined by $\left([x_{12}]\right)=[x_4]$ , where $[u_n]$ denotes the lass of the integer $u$ in $Z_n$ .

Then, for any $[a_{12}],[b_{12}]\in Z_{12}$ , we have

$f\left([a_{12}]+[b_{12}]\right)=f\left([a+b]_{12}\right) \\ \\ =[a+b]_4=[a]_4+[b]_4=f\left([a]_{12}\right)+f\left([b]_{12}\right)$

and

$f\left([a_{12}][b_{12}]\right)=f\left([ab]_{12}\right) \\ \\ =[ab]_4=[a]_4[b]_4=f\left([a]_{12}\right)f\left([b]_{12}\right)$

Therefore, the function is a homomorphism.

7 0

3 years ago

What is the value of the expression?

mr Goodwill [35]

Answer: B.792

Step-by-step explanation:

5 0

1 year ago

Suppose you asked 100 commuters how much they spend each year and obtained a mean of $167 spent on transportation and a standard

Lena [83]

Answer:

The correct answer is "$159 and $175".

Step-by-step explanation:

The give values are:

Mean,

= $167

Standard deviation,

$\sigma$ = $40

Number of commuters,

n = 100

Now,

⇒ $SE=\frac{\sigma}{\sqrt{n} }$

On putting the given values, we get

⇒ $=\frac{40}{\sqrt{100} }$

⇒ $=\frac{40}{10}$

⇒ $=4$

By using the 2 SE rule of thumb, we get

= $$(167 - 2\times 4)$

= $167-8$

= $159$ ($)

Or,

= $(167 + 2\times 4)$

= $167+8$

= $175$ ($)

i.e,

$159 and $175

6 0

2 years ago

Hey! Please, I REALLY need this. Thanks for helping!

shtirl [24]

Answer:

I wont be able to graph it for you, best way is to use desmos or wolfram alpha. This is very easy but since i cant draw it would be difficult to explain. Sorry.

Step-by-step explanation:

6 0

3 years ago

What is the area of this polygon?<br><br> Enter your answer in the box.<br><br> units²

leva [86]

Answer:

45 sq. units

Step-by-step explanation:

We can draw a segment from F to S, splitting this figure into a rectangle and a triangle.

The area of a rectangle is given by the formula A=lw, where l is the length and w is the width.

The width of this figure is the distance of FW; this is 2. The length of this figure is the distance of WC; this is 9. This makes the area of the rectangle A=2(9) = 18 sq. units.

The area of a triangle is given by the formula A=1/2(b)(h), where b is the base and h is the height.

The base of the triangle is given by the distance of FS; this is 9. The height of the triangle is given from N to segment FS; this is 6. This makes the area of the triangle A=1/2(6)(9) = 27 sq. units.

This makes the total area 18+27 = 45 sq. units.

4 0

3 years ago

Read 2 more answers