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2 years ago

16. Consider any eight points such that no three are collinear.

1 answer:

8 0

Let's assume that a,b&c are in one straight line, so cannot form a triangle with each other. Now, total possible Triangle that can be formed choosing any 3 points without any colinear constraint is 8C3 = 56

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Determine if the given mapping phi is a homomorphism on the given groups. If so, identify its kernel and whether or not the mapp

shtirl [24]

Answer:

(a) No. (b)Yes. (c)Yes. (d)Yes.

Step-by-step explanation:

(a) If $\phi: G \longrightarrow G$ is an homomorphism, then it must hold

that $b^{-1}a^{-1}=(ab)^{-1}=\phi(ab)=\phi(a)\phi(b)=a^{-1}b^{-1}$ ,

but the last statement is true if and only if G is abelian.

(b) Since G is abelian, it holds that

$\phi(a)\phi(b)=a^nb^n=(ab)^{n}=\phi(ab)$

which tells us that $\phi$ is a homorphism. The kernel of $\phi$

is the set of elements g in G such that $g^{n}=1$ . However,

$\phi$ is not necessarily 1-1 or onto, if $G=\mathbb{Z}_6$ and

n=3, we have

$kern(\phi)=\{0,2,4\} \quad \text{and} \quad\\\\Im(\phi)=\{0,3\}$

(c) If $z_1,z_2 \in \mathbb{C}^{\times}$ remeber that

$|z_1 \cdot z_2|=|z_1|\cdot|z_2|$ , which tells us that $\phi$ is a

homomorphism. In this case

$kern(\phi)=\{\quad z\in\mathbb{C} \quad | \quad |z|=1 \}$ , if we write a

complex number as $z=x+iy$ , then $|z|=x^2+y^2$ , which tells

us that $kern(\phi)$ is the unit circle. Moreover, since

$kern(\phi) \neq \{1\}$ the mapping is not 1-1, also if we take a negative

real number, it is not in the image of $\phi$ , which tells us that

$\phi$ is not surjective.

(d) Remember that $e^{ix}=\cos(x)+i\sin(x)$ , using this, it holds that

$\phi(x+y)=e^{i(x+y)}=e^{ix}e^{iy}=\phi(x)\phi(x)$

which tells us that $\phi$ is a homomorphism. By computing we see

that $kern(\phi)=\{2 \pi n| \quad n \in \mathbb{Z} \}$ and

$Im(\phi)$ is the unit circle, hence $\phi$ is neither injective nor

surjective.

7 0

3 years ago

A circular flower bed is 24m in diameter and has a circular sidewalk around it that is 2m wide. Find the area of the sidewalk in

sp2606 [1]

Answer:

163.3m²

Step-by-step explanation:

area = Pi(r²)

r = d / 2

r = 12

12² = 144

144 x 3.14 = 452.16

diameter of bigger circle = 24 + 4 (2 on each side)

28 / 2 = 14

14² = 196

196 x 3.14 = 615.44

sidewalk = bigger - smaller

sidewalk = 615.44 - 452.16

sidewalk = 163.3

4 0

1 year ago

1,664 flound ounces = gallons

finlep [7]

Answer:

1,664 fluid ounces = 13 gallons

Step-by-step explanation:

8 0

2 years ago

Consider the division of 15h2 + 10h + 25 by 5h.

Studentka2010 [4]

(15h^2+10h+25)/(5h)
(15h^2+10h+25)/(5)
(3h^2+2h+5)/(h)
(15h^2+10h+25)/(5h)
(5h)(3h)=15h^2
(10h+25)/(5h)
(5h)(2)=10h
(25)/(5h)=5/h=\=
(15h^2+10h+25)/(5h)= 3h+2, with a remainder of 25

7 0

3 years ago

Read 2 more answers

Please help and show work

trasher [3.6K]

Answer:

good morning ❤

Step-by-step explanation:

Miko’s family drove at a constant speed as they left home and traveled to the restaurant where they stopped for lunch. Does the relationship between hours that past x and the distance traveled y represent a proportional relationship? Explain why or why not?

7 0

2 years ago