Determine,in each of the following cases, whether the described system is or not a group. Explain your answers. Determine what is an Abelian group.

a) G = {set of integers , a ∗ b = a − b}

b) G = {set of matrices of size 2 × 2, A ∗ B = A · B}

c) G = {a0, a1, a2, a3, a4ai ∗aj = aIi+jI , if i+j < 5, ai ∗aj = aIi+j-5I , if i+j ≥ 5}

Answer:

Step-by-step explanation:

Given that there are 20 non ionized containers and 73 ionized containers

Two samples are drawn without replacement

a) the probability that the first one selected is not ionized=[tex]\frac{20}{93} =0.215[/tex]

b)  the probability that the second one selected is not ionized given that the first one was ionized

= When first one was ionized we got left over as 20 and 72

Hence = [tex]\frac{20}{92} =0.217[/tex]

c) If with replacement left over 20 and 73 and hence prob = 0.215 as in part a

d) the probability that both are ionized=[tex]\frac{73C2}{93C2} =0.614[/tex]

Answer:

x+10

Step-by-step explanation:

An algebraic expression is an expression which consist of variables(whose values are not fixed like in the form of x,y,a,...), the constants and operators (like +,×,±,-,≤,≥,=,...).

Now for this question we have to give a an algebraic expression for 10 more than a number.

Let the number be x.

We have to show a relation of 10 more than the number. Thus are algebraic expression is of the form x+10.

Algebraic expression: x+10

where x is our variable

+ is our operator and

10 is a constant.

Answer:

They collected $12.73 in all.

Step-by-step explanation:

This problem can be solved by a simple system of equations.

I am going to say that x is the quantity that Travis collects, y the quantity that Jessica collects and z the quantity that Robin collects.

The problems asks how much money did they collect in all?

So [tex]T = x + y + z[/tex]

Solution

The problem states that Travis wants to collect 20% more than Jessica, so:

[tex]100%x = (100%+ 20%)y[/tex]

[tex]100%x = 120%y[/tex]

[tex]x = 1.2y[/tex]

Robin wants to collect 35% more than Travis, so:

[tex]100%z = (100%+35%)x[/tex]

[tex]100%z = 135%x[/tex]

[tex]z = 1.35x[/tex]

Travis collects $4, so [tex]x = 4[/tex]. So:

[tex]x = 1.2y[/tex]

[tex]1.2y = 4[/tex]

[tex]y = \frac{4}{1.2}[/tex]

[tex]y = 3.33[/tex]

------------

[tex]z = 1.35x = 1.35(4) = 5.40[/tex]

The total is:

[tex]T = x + y + z = 4 + 3.33 + 5.40 = $12.73[/tex]

They collected $12.73 in all.

Answer:

1.No, because inverse does not exist.

2.No, because inverse does not exist.

Step-by-step explanation:

We are given that X be  a set and let P(X) be the power set of X.

a. We have to tell P(X) with binary operation

A*B=[tex]A\cap B[/tex] form  a group.

Suppose, x={1,2}

P(X)={[tex]\phi [/tex],{1},{2},{1,2}}

1.Closure  property:[tex]A\cap B\in P(X)[/tex]

{1}[tex]\cap[/tex] {2}=[tex]\phi \in P(X)[/tex]

It is satisfied for all [tex]A,B\in P(X)[/tex]

2.Associative property:[tex](A\cap B)\cap C=A\cap (B\cap C)[/tex]

If A={1},B={2},C={1,2}

[tex]A\cap(B\cap C)[/tex]={1}[tex]\cap[/tex]({2}[tex]\cap[/tex]{1,2})={1}[tex]\cap[/tex] {2}=[tex]\phi[/tex]

[tex](A\cap B)\cap C[/tex]=({1}[tex]\cap[/tex]{2})[tex]\cap[/tex]{1,2}=[tex]\phi\cap[/tex]{1,2}=[tex]\phi[/tex]

Hence, P(X) satisfied the associative property.

3.Identity :[tex]A\cap B=A[/tex] Where B is identity element of P(X)

[tex]A\cap X=A[/tex]

It is satisfied for every element A in P(X).

Hence, X is identity element in  P(X)

4.Inverse :[tex]A\cap B=X[/tex] Where B is an inverse element of A in P(x)

It can not be possible for every element that satisfied [tex]A\cap B=X[/tex]

Hence, inverse does not exist.

Therefore, P(X) is not a  group w.r.t to given binary operation.

2.We have to tell P(X) with the binary operation

A*B=[tex]A\cup B[/tex] form a group

Similarly,

For set X={1,2}

P(X)={[tex]\phi [/tex],{1},{2},{1,2}}

1.Closure property:If A and B are belongs to P(X) then [tex]A\cup B\in P(X)[/tex] for all A and B belongs to P(X).

2.Associative property:[tex]A\cup (B\cup C)=(A\cup B)\cup C[/tex]

If A={1},B={2},C{1,2}

[tex]A\cup B[/tex]={1}[tex]\cup [/tex]{2}={1,2}

[tex](A\cup B)\cup C[/tex]={1,2}[tex]\cup [/tex]{1,2}={1,2}

[tex]B\cup C[/tex]={2}[tex]\cup [/tex]{1,2}={1,2}

[tex]A\cup (B\cup C)[/tex]={1}[tex]\cup [/tex]{1,2}={1,2}

Hence, P(X) satisfied the associative property.

3.Identity :[tex]A\cup B=A[/tex] Where B is identity element of P(X)

Only [tex]\phi[/tex] is that element for every A in P(X) that satisfied [tex]A\cup B=A[/tex]

Hence, [tex]\phi[/tex] is identity element of P(X) w.r.t union.

4.Inverse element :

[tex]A\cup B=\phi[/tex] where B is  an inverse element of A in P(X)

It is not possible for every element that satisfied the property.

Hence, inverse does not exist for each element in P(X).

Therefore, P(X) is not  a group w.r.t binary operation.

Final answer:

The power set P(X) does not form a group with the intersection operation because there are no inverse elements for all elements of P(X). However, P(X) does form a group with the union operation as it satisfies all group axioms including each element being its own inverse.

Explanation:

Power Set Operations as Groups

To determine if P(X) forms a group with the specified operations, we need to check if the operations satisfy the group axioms: closure, associativity, identity, and invertibility.

a) Intersection Operation *

For the operation A * B = A ∩ B (intersection), all subsets of X including X are closed under intersection, and intersection is associative. The set X itself acts as an identity element because A ∩ X = A for any A in P(X). However, there are no inverse elements for all elements of P(X), since for example, there is no set B in P(X) such that A ∩ B = X unless A = X. Therefore, P(X) with intersection does not form a group.

b) Union Operation *

For the operation A * B = A ∪ B (union), all subsets of X including the empty set are closed under union, and union is associative. The empty set ∅ acts as an identity because A ∪ ∅ = A for any A in P(X). Every element is its own inverse since A ∪ A = ∅. Hence, P(X) with the union operation does form a group.

Answer:

standard  deviation is 146 cm

Computed value of standard deviation is not near to original value.

Step-by-step explanation:

Given data:

n is number of brains = 50

low volume  = 904 cm

high volume of brain = 1488 cm

As we know that range is 4 times the standard deviation so we have[tex]Range = 4\times standard\ deviation[/tex]

R = HIGH -  LOW

  = 1488 - 904

  = 584

Therefore we have

standard deviation[tex] = \frac{R}{4}[/tex]

                                [tex]= \frac{584}{4}[/tex]

standard  deviation is 146 cm

Original deviation is given as 175.5 cm

Computed value of standard deviation is not near to original value.

To construct a matrix A in which a vector b is not spanned by A's columns, choose b to be a vector not obtainable by linear combinations of A's columns. For example, if A is a 3 x 3 matrix with consecutive integers, then b with a sufficiently different third entry would not be in the span of A's columns.

To construct a 3 x 3 matrix A with non-zero entries where vector b in R3 is not in the set spanned by the columns of A, it is essential to ensure that b is not a linear combination of the columns of A. First, let's define matrix A with arbitrary non-zero entries:

A = 'p'\n'\n'[1 2 3]\n'\n'[4 5 6]\n'\n'[7 8 9]\n

To find a vector b that is not spanned by the columns of A, we should choose a vector that is not a linear combination of these columns. For instance:

b = 'p'\n'\n'[1]\n'\n'[1]\n'\n'[10]\n

This vector b cannot be formed by any combination of the columns of matrix A, because there's no scalar multiples that we can multiply the columns of A by to get a z-component of 10 while also having the x and y components equal to 1. In other words, b is outside the column space of A, showing that linear independence is not achieved.

The vector [tex]\( \mathbf{b} \)[/tex] is clearly not a multiple of [tex]\( \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \)[/tex]. he matrix  [tex]\[ A = \begin{bmatrix} 1 2 3 \\ 2 4 6 \\ 3 6 9 \\ \end{bmatrix} \][/tex] and vector [tex]\( \mathbf{b} \)[/tex] provided above satisfy the given conditions.

To construct a 3 x 3 matrix [tex]\( A \)[/tex] with nonzero entries and a vector [tex]\( \mathbf{b} \)[/tex] in [tex]\( \mathbb{R}^3 \)[/tex]  such that  is not in the [tex]\( \mathbf{b} \)[/tex] span of the columns of [tex]\( A \)[/tex], we need to ensure that [tex]\( A \)[/tex] is not full rank. A matrix [tex]\( A \)[/tex]  is full rank if its columns are linearly independent and span[tex]\( \mathbb{R}^3 \)[/tex]. Since we want [tex]\( \mathbf{b} \)[/tex] not to be in the span of [tex]\( A \)[/tex] , [tex]\( A \)[/tex] must have a rank less than 3.

Let's construct [tex]\( A \)[/tex] such that two of its columns are multiples of each other, which will ensure that the matrix is rank-deficient (rank less than 3). For example:

[tex]\[ A = \begin{bmatrix} 1 2 3 \\ 2 4 6 \\ 3 6 9 \\ \end{bmatrix} \][/tex]

Here, the second column is twice the first column, and the third column is three times the first column. This implies that the columns of [tex]\( A \)[/tex] are linearly dependent, and the rank of [tex]\( A \)[/tex] is 1.

Now, let's choose a vector [tex]\( \mathbf{b} \)[/tex] that is not a multiple of the first column of [tex]\( A \)[/tex]. For instance: [tex]\[ \mathbf{b} = \begin{bmatrix} 1 \\ 0 \\ 0 \\ \end{bmatrix} \][/tex]

The vector [tex]\( \mathbf{b} \)[/tex] is clearly not a multiple of [tex]\( \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \)[/tex], which is the first column of [tex]\( A \)[/tex]. Therefore, [tex]\( \mathbf{b} \)[/tex] cannot be written as a linear combination of the columns of [tex]\( A \)[/tex] , and thus it is not in the span of [tex]\( A \)[/tex].

To verify this, we would typically attempt to solve the system [tex]\( A\mathbf{x} = \mathbf{b} \)[/tex] for [tex]\( \mathbf{x} \)[/tex]. If there is no solution, then [tex]\( \mathbf{b} \)[/tex] is not in the span of the columns of [tex]\( A \)[/tex]. In this case, since  [tex]\( A \)[/tex]  is rank-deficient and [tex]\( \mathbf{b} \)[/tex] is not a multiple of any column of [tex]\( A \)[/tex] , the system has no solution.

In conclusion, the matrix [tex]\( A \)[/tex] and vector [tex]\( \mathbf{b} \)[/tex] provided above satisfy the given conditions.

Answer:

Where Z is the exhaustive multitude of the whole numbers

[tex]\text{Hello there!}\\\\\text{If you're solving for z:}\\\\z-4=10\\\\\text{Add 4 to both sides}\\\\z=14\\\\\text{Your answer would be:}\,\boxed{z=14}[/tex]

A population is the entire group of interest in a researcher's study, while a sample is a subset of this group from which data is collected. The aim is for the sample to be representative of the population to accurately draw generalizations. Effective sampling strategies and recruitment are vital for this representation.

To understand the concepts of population and sample in the context of statistics, we can differentiate between the two. Population refers to the entire group that is the focus of a researcher's study, which can be a broad group, like all adults over the age of 18 in the United States, or more specific, such as 'mid-season maturity corn plants on irrigated farms near Grand Island, Nebraska.'

A sample, on the other hand, is a subset of the population from which researchers actually collect data. It represents a smaller group selected to draw conclusions about the population. The validity of these conclusions often depends on how well the sample represents the population, aiming for the sample characteristics to match those of the population. For instance, if surveying the relative proportion of cars to trucks driving down a street, a sample observed during an uncharacteristic time of day may not provide a representative view of the overall traffic pattern.

In research, sampling strategies and recruitment techniques are important to ensure that the sample accurately reflects the population. For example, choosing individuals to participate in a study so that each has a known chance of being included makes for a better representation of the population. Researchers then analyze the sample data and attempt to generalize their findings to the entire population.

Answer: Blood type will be A when event "A" happened and event "B" did not happen. Blood type will be B when event "A" did not happened and event "B" happened. Blood type will be AB when both events happened and blood type will be O when both events did not happen.

Step-by-step explanation:

S={AntiA reacts; AntiA does not react; AntiB reacts; AntiB does not react}

If AntiA reacts and AntiB reacts = AB (A∩B)

If AntiA does not react and AntiB does not react= O (A'∩B')

If AntiA reacts and AntiB does not react= A (A∩B')

If AntiA does not react and AntiB reacts= B (A'∩B)

The blood type is determined by observing the blood reaction to anti-A and anti-B antigens. Type A reacts with anti-A, Type B reacts with anti-B, Type AB reacts with both, and Type O doesn't react with either.

The process of identifying a person's blood type using anti-A and anti-B antigens is straightforward. If the person's blood agglutinates (clumps together) when anti-A antigens are added, it means the blood has type A glycoproteins on the surface and the person has type A or AB blood. This is what we call event A.

Similarly, if the blood reacts with the anti-B antigen (event B), it means the person has type B or AB blood. If the blood reacts to both anti-A and anti-B antigens, it must be type AB. If the blood doesn't react with either antigen (the complement of both A and B events), it signifies the person has type O blood, which lacks both A and B glycoproteins on the erythrocyte surfaces.

It's also worth noting that AB blood can accept blood from any type (universal acceptor), while O blood type can be transferred to any blood type (universal donor) as it doesn't cause an immune response due to lack of A and B antigens.

https://brainly.com/question/34140937

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Answer:

  19 days

Step-by-step explanation:

Since Ali gained 3 days each year, she has gained ...

  (3 days/yr)×(5 yr) = 15 days

Added to the 4 days she started with, her vacation time is now ...

  4 days + 15 days = 19 days

After 5 years of employment at the fashion magazine, Ali has 19 days of paid vacation time.

Ali, after working for a fashion magazine for 5 years, will have a certain number of paid vacation days accumulated. She gets 4 days of paid vacation initially. For every year she works, she gains another 3 days of vacation. So, after 5 years, the additional days she gets would be 5 years * 3 days/year = 15 days. Adding this to her initial 4 days, Ali gets 15 + 4 = 19 days of paid vacation. But her maximum limit is 26 days. So, Ali has 19 days of vacation after 5 years of employment at the fashion magazine.

https://brainly.com/question/33104803

Final answer:

To find the equation of a line that passes through a given point and has a given x-intercept, we can use the point-slope form of a line.

Explanation:

To find the equation of a line that passes through the point (-8, 4) and has an x-intercept of -10, we can use the slope-intercept form of a line, which is y = mx + b.

First, let's find the slope of the line using the given information. The x-intercept represents the point where the line crosses the x-axis, so if the x-intercept is -10, we know that the point (-10, 0) is on the line.

Using the formula for slope, which is m = (y2 - y1) / (x2 - x1), we can calculate the slope of the line as (0 - 4) / (-10 - (-8)) = -4 / -2 = 2.

Now, we can use the point-slope form of a line, which is y - y1 = m(x - x1), where (x1, y1) is a point on the line.

Substituting the values (-8, 4) and m = 2 into the equation, we have y - 4 = 2(x - (-8)).

Simplifying the equation, we get y - 4 = 2x + 16.

Finally, isolating y, we arrive at the equation of the line: y = 2x + 20.

Answer:

the strand contains 28% of adenine.

Step-by-step explanation:

We have only four components, and we only know one of them:

[tex]\left[\begin{array}{cc}C&?\\G&22\%\\A&?\\T&?\end{array}\right][/tex]

Cytosine has a relation 1 to 1 with G, therefore the strand must contain the same amount of C as it posses G:

[tex]\left[\begin{array}{cc}C&22\%\\G&22\%\\A&?\\T&?\end{array}\right][/tex]

Therefore:

[tex]A + T = 100\% - (C+G)[/tex]

This is because the strain only contains those 4 components.

since A and T have also a 1 to 1 relation, we can state that A = T in quantity.

So:

[tex]A + A = 100\% - (C+G)[/tex]

[tex] 2A = 100\% - (22\%+22\%)[/tex]

[tex]A = \frac{100\%-44\%}{2}[/tex]

[tex]A = \frac{56%}{2}[/tex]

[tex] A = 28\%[/tex]

Answer: A. 10% of the time, the population parameter will lie outside of the interval.

Step-by-step explanation:

If we have [tex]b\%[/tex] confidence interval is that we are [tex]b\%[/tex] certain that it contains the true population parameter in it.

Similarly , if  we have a 90% confidence interval for a population parameter, then we are 90% certain that it contains the true population parameter in it.

i.e. 10% not certain that it contains the true population parameter in it.

i.e. 10% of the time, the population parameter will lie outside of the interval.

Answer:

Since y is a solution of the homogeneus system then satisfies Ay=0.

Since z is a solution of the system Ax=b then satisfies Az=b.

Now, we will show that A(y+z)=b.

Observe that A(y+z)=Ay+Az by properties of the product of matrices.

By hypotesis Ay=0 and Az=b.

Then A(y+z)=Ay+Az=0+b=b.

Then A(y+z)=b, this show that y+z is a solution of the system Ax=b.

Answer:

Part 1)

Projection of vector A on vector B equals 19 units

Part 2)

Projection of vector B' on vector A' equals 35 units

Step-by-step explanation:

For 2 vectors A and B the projection of A on B is given by the vector dot product of vector A and B

Given

[tex]\overrightarrow{v_{a}}=\widehat{i}-2\widehat{j}+\widehat{k}[/tex]

Similarly vector B is written as

[tex]\overrightarrow{v_{b}}=4\widehat{i}-4\widehat{j}+7\widehat{k}[/tex]

Thus the vector dot product of the 2 vectors is obtained as

[tex]\overrightarrow{v_{a}}\cdot \overrightarrow{v_{b}}=(\widehat{i}-2\widehat{j}+\widehat{k})\cdot (4\widehat{i}-4\widehat{j}+7\widehat{k})\\\\\overrightarrow{v_{a}}\cdot \overrightarrow{v_{b}}=1\cdot 4+2\cdot 4+1\cdot 7=19[/tex]

Part 2)

Given vector A' as

[tex]\overrightarrow{v_{a'}}=2\widehat{i}+3\widehat{j}+6\widehat{k}[/tex]

Similarly vector B' is written as

[tex]\overrightarrow{v_{b'}}=\widehat{i}+5\widehat{j}+3\widehat{k}[/tex]

Thus the vector dot product of the 2 vectors is obtained as

[tex]\overrightarrow{v_{b'}}\cdot \overrightarrow{v_{a'}}=(\widehat{i}+5\widehat{j}+3\widehat{k})\cdot (2\widehat{i}+3\widehat{j}+6\widehat{k})\\\\\overrightarrow{v_{a'}}\cdot \overrightarrow{v_{b'}}=1\cdot 2+5\cdot 3+3\cdot 6=35[/tex]

Answer: P=8

Step-by-step explanation:

3p-5=19

U turn -5 to +5

Then u add 5 to both sides, -5 and 19

3p-5=19

+5=+5

-5 and +5 cancel each other out so know it’s

3p=24

Because 19 plus 5 is 24

Now u have to get the variable by itself by dividing 3 on both sides of the equal sign 3p and 24

3 and 3 cancel each other out so now you only have p=24 but then 24 divided 3 is 8

Answer:

"There exists a good book that does not have a plot twist and does not have a surprise ending".

Step-by-step explanation:

We negate the universal quantifier "for all" or equivalently "In every" using the existential quantifier "There exists". So, we negate "In every good book" as "There exists a good book". In the other hand, we have the propositions

P: there is a plot twist

Q: there is a surprise ending,

and the conjunction

P ∨ Q. We negate this conjunction using the De Morgan's Laws as

¬(P∨Q) = ¬P∧¬Q

i.e., does not have a plot twist and does not have a surprise ending. Therefore, we negate "In every good book there is a plot twist or surprise ending" as "There exists a good book that does not have a plot twist and does not have a surprise ending".

Answer:

(a) 84.2

(b) 10.6

Step-by-step explanation:

To solve this questions we can use the standardization formula, where we know that if [tex]X\sim N(\mu,\sigma^2)[/tex] then [tex]Z=\frac{X-\mu}{\sigma} \sim N(0,1)[/tex]

So for

(a) we know that the z score for the 70th percentile is 0.524, so using the normalization equation we have

[tex]\frac{X-\mu}{\sigma}=0.524[/tex]

[tex]X=0.524*8+80=84.192[/tex]

(b) We can procede as above and get

[tex]P(X<70)=P(\frac{X-80}{8}<\frac{70-80}{8})=P(Z<-1.25)=0.1056[/tex]

Answer:

|-3| = 3. It indicates the distance the number -3 is from zero

Step-by-step explanation:

We have been asked that:

Explain the difference in meaning between |-3| and-3

Actually |-3| = 3. It indicates the distance the number -3 is from zero, which is 3 units in this case. Look at the attached picture. If you start from -3. Then you have to walk 3 spaces to get to the number zero. The distance in the attached picture is along a number line....

Answer:

Maximum altitude: 497.96 ft

Horizontal range: 1007.37 ft

Speed at impact:  165.21 ft/s

Step-by-step explanation:

angle(α) = atan (7/6) = 49.4°

Maximum altitude is given by the formula:

[tex]h=\frac{V_0^2sin^2\alpha }{2g}[/tex]

[tex]h=\frac{100^2 sin^2(49.4)}{2*9.81} =\frac{9770}{19.62}=497.96 ft/s[/tex]

Horizontal range is given by the formula:

[tex]X=\frac{V_0^2sin(2\alpha)}{g}[/tex]

[tex]X=\frac{100^2sin(2*49.4)}{*9.81}=1007.37 ft[/tex]

Speed at impact is given by the formula:

[tex]V_f=\sqrt{V_x^2 + Vy^2}[/tex]

where:

[tex]V_x= V_0cos(\alpha )= 100cos(49.4)=65.07 ft/s[/tex]

[tex]V_y=V_0sin(\alpha ) + gt=100sin(49.4)+9.81(t)[/tex]

[tex]t=\frac{V_0sin(\alpha) }{g}=\frac{100sin(49.4)}{9.81}=7.74s[/tex]

So;

[tex]V_y= 100sin(49.4)+(9.81)(7.74)= 151.86 ft/s[/tex]

[tex]Vf=\sqrt{V_x^2 + V_y^2} =\sqrt{65.07^2+151.86^2}=165.21 ft/s[/tex]

Answer:

The answer is 12500....

Step-by-step explanation:

We have been asked that what is the sum of 9260 and 3240?

The sum of two numbers is the result you obtain by adding the two numbers together.

Addition is the mathematical process of putting things together. The plus sign "+" shows  that numbers are added together. We start adding the numbers from right hand side.

We have two values 9260 and 3240. We will add these two values together.

  9  2   6   0

+ 3  2   4   0

__________

 12 5   0   0

Thus the answer is 12500....

Answer:

When x=1 the y-coordinate is 6.

Step-by-step explanation:

The given linear equation is

[tex]3x+2y=15[/tex]

We need to find the y-coordinate when x=1.

Substitute x=1 in the given equation, to find the y-coordinate.

[tex]3(1)+2y=15[/tex]

[tex]3+2y=15[/tex]

Subtract both sides by 3.

[tex]3+2y-3=15-3[/tex]

[tex]2y=12[/tex]

Divide both sides by 2.

[tex]\frac{2y}{2}=\frac{12}{2}[/tex]

[tex]y=6[/tex]

Therefore at x=1 the y-coordinate is 6.

Answer:

The formula is not valid because the commutative property with respect to the matrix product operation is not fulfilled in the vector space of the real matrices.

Step-by-step explanation:

The formula is not valid because the commutative property with respect to the matrix product operation is not fulfilled in the vector space of the real matrices. That is, AB does not necessarily equal BA.

[tex](A+B)(A-B) = A^2-AB+BA-B^2\neq A^2 - B^2[/tex]

[tex]A=\left[\begin{array}{ccc}1&0&0\\0&0&6\\0&8&0\end{array}\right] \\B=\left[\begin{array}{ccc}0&2&0\\6&0&0\\0&0&9\end{array}\right] \\(A -B) = \left[\begin{array}{ccc}1&-2&0\\-6&0&6\\0&8&-9\end{array}\right]\\\\(A + B) = \left[\begin{array}{ccc}1&2&0\\6&0&6\\0&8&9\end{array}\right]\\(A - B)(A + B) = \left[\begin{array}{ccc}-11&2&-12\\-6&36&54\\48&-72&-33\end{array}\right]\\A^2 - B^2 = \left[\begin{array}{ccc}-11&0&0\\0&36&0\\0&0&-33\end{array}\right]\\[/tex]

You can use the fact that multiplication of matrices is dependent on the order of the matrices which are multiplied.

The correct option for the given condition is

Option C:  The formula is not valid because in general, AB ≠ BA for matrices.

It might be that AB = BA for two matrices A and B but it is very rare and thus, cannot be generalized as identity.

Suppose A has got shape (m,n) (m rows, n columns)

and B has got shape (n,k) (n rows, k columns), then AB is defined but BA is not defined if k  ≠  m.

Also, even if k =m, we can't say for sure that AB = BA

Thus, usually we have AB ≠ BA

For two matrices A and B, supposing that AB and BA are defined, then we have

[tex](A+B)(A-B) = A(A-B) + B(A -B) = A^2 -AB + BA - B^2[/tex]

Since may or may not have AB equal to BA, thus, we cannot cancel those two middle terms to make 0 matrix.

Thus,

The correct option for the given condition is

Option C:  The formula is not valid because in general, AB ≠ BA for matrices.

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Answer and Explanation:

Given : [tex]P(n): 1 + 2 + 2^2 + 2^3 + . . . + 2^n = 2^{n+1} - 1[/tex], n=0,1,2,..

To find : The values of following expression ?

Solution :

The function is [tex]P(n)=2^{n+1} - 1[/tex]

1) Value of P(0),

[tex]P(0)=2^{0+1} - 1[/tex]

[tex]P(0)=2^{1} - 1[/tex]

[tex]P(0)=2 - 1[/tex]

[tex]P(0)=1[/tex]

2) Value of P(1),

[tex]P(1)=2^{1+1} - 1[/tex]

[tex]P(1)=2^{2} - 1[/tex]

[tex]P(1)=4- 1[/tex]

[tex]P(1)=3[/tex]

3) Value of P(2),

[tex]P(2)=2^{2+1} - 1[/tex]

[tex]P(2)=2^{3} - 1[/tex]

[tex]P(2)=8- 1[/tex]

[tex]P(2)=7[/tex]

4) Value of P(n+1),

[tex]P(n+1)=2^{n+1+1} - 1[/tex]

[tex]P(n+1)=2^{n+2} - 1[/tex]

Answer:

[tex]1.167 km[/tex]

Step-by-step explanation:

We are given with-

Height of airplane from water [tex]a[/tex] = [tex]425 m[/tex]

Angle of depression (∅)= 20°

Now,

[tex]tan(20) = \frac{b}{a}[/tex]

[tex]a = \frac{b}{tan(20)} \\a = \frac{425}{tan(20} \\a = 1167.677 m[/tex]

[tex]a = 1.167 km[/tex]

Answer:

2000ml = 2L of this IV fluid should be infused.

Step-by-step explanation:

This problem can be solved by a simple rule of three, in which the relationship between the measures is direct, which means that there is a cross multiplication.

The problem states that each 75 mg of the medication contains 500 ml of IV fluid. How many ml of IV fluid are there in 300 mg of the medication?

So

75mg - 500ml

300 mg - x ml

[tex]75x = 300*500[/tex]

[tex]x = \frac{300*500}{75}[/tex]

[tex]x = 4*500[/tex]

[tex]x = 2000[/tex]ml

2000ml = 2L of this IV fluid should be infused.

To administer 300 mg of Drug B, given that 75 mg is in 500 ml of IV fluid, you need infuse 2000 ml of the IV fluid. This was calculated using proportional relationships based on the concentration of the drug in the fluid.

To determine how much IV fluid should be infused to provide the patient with 300 mg of Drug B, we can use a simple proportion based on the concentration of the drug in the IV fluid.

0.15 mg/ml = 300 mg / X ml

Solving for X, we get:

X = 300 mg / 0.15 mg/ml

X = 2000 ml

Thus, 2000 ml of the IV fluid should be infused to provide the patient with 300 mg of Drug B.

Answer:

4x² + 2x + 6 = 0

Step-by-step explanation:

The polynomial which has highest degree 2 is known as quadratic polynomial. It is of the form:

ax² + bx + c = 0

where, a ≠ 0 and a, b & c are any constant.

We have given three points (-1, 8), (0, 6), and (2, 26)

Putting these value of (x, y) in quadratic equation one by one.

We get, three equations:

8 = a - b + c

6 = c

26 = 4a + 2b + c

Solving these equations, We get,

a = 4, b = 2 and c = 6

Now putting these values of a, b, and c in standard quadratic equation.

We get,

4x² + 2x + 6 = 0

which is required equation.

Answer:

The Venn diagram has 8 different colored areas, in the image attached you can see the colors and the sets that make up the Venn diagram:

1. white:

U / (A ∪ B ∪ C)

2. black:

A ∩ B ∩ C

3. yellow:

A / (A ∩ B) ∪ (A ∩ C)

4. blue:

B / (A ∩ B) ∪ (B ∩ C)

5. red:

C / (B ∩ C) ∪ (A ∩ C)

6. green:

A ∩ B / (A ∩ B ∩ C)

7. orange:

A ∩ C / (A ∩ B ∩ C)

8. violet:

B ∩ C / (A ∩ B ∩ C)

Step-by-step explanation:

Each set has a color, A is yellow, B blue and C red. Taking the notation of sets and the law of combining colors, you can find all the colors that make up the diagram.  

1. white: the universal set (U) has all the elements, except for those that are not in the A, B and C sets.

U / (A ∪ B ∪ C)

2. black: this color is formed with the combination of all colors in the diagram, and it contains the intersection of the 3 sets.

A ∩ B ∩ C

For colors yellow, blue and red you can take each set A, B and C and subtract from each one of them the union of the intersection of the other two sets.

3. yellow:

A / (A ∩ B) ∪ (A ∩ C)

4. blue:

B / (A ∩ B) ∪ (B ∩ C)

5. red:

C / (B ∩ C) ∪ (A ∩ C)

Finally, for colors green, orange and violet you take the intersection of each set A ∩ B, A ∩ C and B ∩ C and subtract from them the elements in the black set.

6. green:

A ∩ B / (A ∩ B ∩ C)

7. orange:

A ∩ C / (A ∩ B ∩ C)

8. violet:

B ∩ C / (A ∩ B ∩ C)

The given Venn diagram, with three sets A, B, and C and given intersections, would create 8 different colored areas, including the universal set color.

The question is about understanding the structure and representation of a Venn diagram. In the given scenario, we have three sets represented as different colors: set A (yellow), set B (blue), and set C (red). Our universal set is white. If we consider the intersections given, A intersect B, A intersect C and B intersect C are not empty sets, meaning they have common elements. Hence, in a Venn diagram, such intersections will create separate areas.

The key component in a Venn diagram is color representation. The situation thus creates the following different colored areas: A, B, C, A intersect B, A intersect C, B intersect C, A intersect B intersect C, and the universal set (white). Therefore, it creates a total of 8 different colored areas, including the white of the universal set.

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Answer:

  5/2

Step-by-step explanation:

You want to find x when f(x) = 4

  4 = 2x -1

  5 = 2x . . . . . add 1

  5/2 = x . . . . . divide by 2

The input corresponding to an output of 4 is 5/2.

Answer:

0, 1 or 2.

Step-by-step explanation:

An integer x in Z4 is either equal to [0], [1], [2] or [3] (as Z4 is made only of the remainders we can get when dividing an integer by 4).

If x was equal to [0] in Z4, then x^2 = [0]*[0]=[0] in Z4.

If x was equal to [1] in Z4, then x^2 = [1]*[1]=[1] in Z4.

If x was equal to [2] in Z4, then x^2 = [2]*[2]=[4]=[0] in Z4 (as 4 and 0 are the same in Z4, given that both numbers leave a remainder of 0 when divided by 4).

If x was equal to [3] in Z4, then x^2 = [3]*[3]=[9]=[1] in Z4 (as 9 and 1 are the same in Z4, given that both numbers leave a remainder of 1 when divided by 4).

Therefore, in Z4 x^2+y^2 is either a sum of the form [0]+[0], or [0]+[1], or [1]+[0], or [1]+[1], which means we can only get either [0], [1] or [2].