To prove that C is a field, we need to show that it satisfies the following 11 axioms of a field:
C is closed under addition: For all a, b ∈ C, a + b ∈ C.
C is closed under multiplication: For all a, b ∈ C, ab ∈ C.
C is associative under addition: For all a, b, c ∈ C, (a + b) + c = a + (b + c).
C is associative under multiplication: For all a, b, c ∈ C, (ab)c = a(bc).
C has an additive identity: There exists 0 ∈ C such that for all a ∈ C, a + 0 = a.
C has a multiplicative identity: There exists 1 ∈ C such that for all a ∈ C, a1 = a.
C has additive inverses: For all a ∈ C, there exists b ∈ C such that a + b = 0.
C has multiplicative inverses: For all a ≠ 0 ∈ C, there exists b ∈ C such that ab = 1.
C is commutative under addition: For all a, b ∈ C, a + b = b + a.
C is commutative under multiplication: For all a, b ∈ C, ab = ba.
Multiplication distributes over addition: For all a, b, c ∈ C, a(b + c) = ab + ac.
Proofs of the 11 axioms:
To show that C is closed under addition, we can use the fact that the sum of two complex numbers is also a complex number. That is, for any a, b ∈ C, a + b = (a + bi) + (b + di)i, where i is the imaginary unit. This is clearly a complex number, so C is closed under addition.
To show that C is closed under multiplication, we can use the fact that the product of two complex numbers is also a complex number. That is, for any a, b ∈ C, ab = (a + bi)(b + di) = (ab - bd) + (ad + bc)i, which is also clearly a complex number, so C is closed under multiplication.
To show that C is associative under addition, we can use the associative property of addition for real numbers. That is, for any a, b, c ∈ C, (a + b) + c = a + (b + c).
To show that C is associative under multiplication, we can use the associative property of multiplication for real numbers. That is, for any a, b, c ∈ C, (ab)c = a(bc).
To show that C has an additive identity, we can use the fact that the real number 0 is also a complex number, and that for any a ∈ C, a + 0 = a.
To show that C has a multiplicative identity, we can use the fact that the real number 1 is also a complex number, and that for any a ∈ C, a1 = a.
To show that C has additive inverses, we can use the fact that for any a ∈ C, there exists a complex number -a such that a + (-a) = 0. This is because the real numbers have additive inverses, and the imaginary unit i has an additive inverse -i.
To show that C has multiplicative inverses, we can use the fact that for any a ≠ 0 ∈ C, there exists a complex number 1/a such that a(1/a) =
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The function f(x) is shown on the graph.
The graph shows a downward opening parabola with a vertex at 3 comma 25, a point at negative 2 comma 0, a point at 8 comma 0, a point at 0 comma 16, and a point at 6 comma 16.
What is the standard form of the equation of f(x)?
f(x) = x2 − 6x + 16
f(x) = x2 + 6x + 16
f(x) = −x2 − 6x + 16
f(x) = −x2 + 6x + 16
Therefore, the equation of the parabola is:
f(x) = -(x - 3)² + 25
f(x) = -x² + 6x + 16
So the answer is f(x) = -x² + 6x + 16.
What does a vertex in mathematics mean?Typically, the intersection of two or more lines or edges forms a vertex, a singular point on a mathematical object. Graphs, polygons, polyhedra, and angles are the shapes that contain vertices most commonly. Nodes are another name for vertices in a graph.
Since the vertex of the parabola is at (3, 25), we know that the equation of the parabola is of the form:
f(x) = a(x - 3)² + 25
where "a" is a constant that determines the shape of the parabola. We also know that the parabola passes through the points (-2, 0), (8, 0), (0, 16), and (6, 16).
Let's plug in the coordinates of one of the points on the parabola to find the value of "a". For example, if we plug in the coordinates of the point (0, 16), we get:
16 = a(0 - 3)² + 25
-9 = 9a
a = -1
Therefore, the equation of the parabola is:
f(x) = -(x - 3)² + 25
f(x) = -x² + 6x + 16
So the answer is f(x) = -x² + 6x + 16.
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Some on pls answer a s a p i need help will give brainlist thingy
Math question 5 help
The steepest road in the world is Canton Avenue in Pittsburgh, Pennsylvania, with a grade of 37%. Grade is defined as the amount of vertical rise (in ft) over 100 ft of horizontal distance (so a road that rises 6 ft over 100 ft of horizontal distance is 6 100 = .06 = 6%). If the 37% grade of Canton Avenue goes for 21 ft of horizontal distance, how much does it rise? What angle does this grade make with the ground?
The steepest road in the world, Canton Avenue in Pittsburgh, Pennsylvania, has a grade of 37%. This means that for every 100 ft of horizontal distance, the road rises 37 ft. To find out how much the road rises for 21 ft of horizontal distance, we can use the formula:
rise = grade × distance
Plugging in the values we have:
rise = 0.37 × 21
rise = 7.77 ft
Therefore, the road rises 7.77 ft for 21 ft of horizontal distance.
To find the angle that this grade makes with the ground, we can use the formula:
tan θ = rise ÷ distance
Plugging in the values we have:
tan θ = 7.77 ÷ 21
tan θ = 0.37
θ = tan^-1(0.37)
θ = 20.3°
Therefore, the grade of Canton Avenue makes an angle of 20.3° with the ground.
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a pair of pants usually sells at 500.00 during a sale , it is being sold at a 40% discount how much is the discount?how much will u pay for a pair of pants during the sale?
0.12x= 1 I need help on this please
Answer:
x=8.3 recurring
Step-by-step explanation:
1÷0.12=8.3 recurring
x=8.3 recurring
or as a fraction 8 1/3
For each of the following, find the formula for an
exponential function that passes through the two points
given.
a. (0,4) and (2,64)
f(x)=?
b. (0,810) and (2,10)
g(x)=?
The formula for an exponential function that passes through the two points are
a. (0,4) and (2,64)
f(x)= 4(4ˣ)
b. (0,810) and (2,10)
g(x)=810(1/9)ˣ
The following is the formula for an exponential function that traverses two points:
f(x) = abˣ
Where a is the initial value and b is the growth rate.
To find the formula for an exponential function that passes through the two points given, we can plug in the values of x and y into the formula and solve for a and b.
For the first set of points, (0,4) and (2,64), we can plug in the values of x and y into the formula and solve for a and b:
4 = ab⁰
64 = ab²
Simplifying the first equation gives us:
a = 4
Substituting this value of a into the second equation gives us:
64 = 4b²
Solving for b gives us:
b = √(64/4) = 4
Therefore, the formula for the exponential function that passes through the two points (0,4) and (2,64) is:
f(x) = 4(4ˣ)
For the second set of points, (0,810) and (2,10), we can plug in the values of x and y into the formula and solve for a and b:
810 = ab⁰
10 = ab²
Simplifying the first equation gives us:
a = 810
Substituting this value of a into the second equation gives us:
10 = 810b²
Solving for b gives us:
b = √(10/810) = 1/9
Therefore, the formula for the exponential function that passes through the two points (0,810) and (2,10) is:
g(x) = 810(1/9)ˣ
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estimate the product of 3.8 and 11.9 by rounding
The product or multiple of 3.8 and 11.9 by rounding off (simplification) is equal to 45.22.
What is simplification in mathematics?Simplification means keeping it simple. In mathematics, it simplifies or simplifies formulas/fractions/problems into simpler forms. Simplify the problem with calculations and solutions.
Simplification generally means finding answers to complex calculations involving division, multiplication, square roots, cube roots, plus and minus numbers.
Why do we use simplification?Simplicity complicates deadpan, making it easier to understand and solve. Here are some advantages of solving a problem or equation by simplification.
It helps you solve your problem in fewer steps. Complex problems can be reduced to simpler forms by following the rules of simplification
According to the question:
3.8×11.9 = (38/10)×(119/10)
= 4522/100
= 45.22
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If the constant of variation is 12 and y = 36 and they vary directly, what does x equal?
If the constant of variation is 12 and y = 36 and they vary directly, then x equals 3.
Direct variation is when two quantities, x and y, are related in such a way that the ratio of their values is always the same. This means that y = kx, where k is the constant of variation.
In this case, we are given that k = 12 and y = 36. We can plug these values into the equation to find x:
36 = 12x
To solve for x, we can divide both sides of the equation by 12:
36/12 = 12x/12
This simplifies to:
3 = x
Therefore, x equals 3.
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solve 2x + 3y = 4 and -x + 4y = -13 algebraically
2x + 3y = 4
-x + 4y = -13 /×2
2x + 3y = 4
-2x + 8y = -26
11y = -22
y = -2
2x + 3(-2) = 4
2x + (-6) = 4
2x = 10
x = 5
check:
2(5) + 3(-2) = 4
10 + (-6) = 4
4 = 4
L = R
-(5) + 4(-2) = -13
-5 + (-8) = -13
-13 = -13
L = R
∴ x = 5, y = -2
Can some one help me with dis math
Answer:
Step-by-step explanation:
c
Answer:E
Step-by-step explanation:
Find the Euclidean inner product of the given vectors. u=[[5],[3],[-4]],v=[[1],[0],[-5]]
The Euclidean inner product of the given vectors is 25.
The Euclidean inner product of two vectors u and v is defined as the sum of the products of the corresponding entries of the vectors. In mathematical terms, it is given by:
Euclidean inner product = u[1]*v[1] + u[2]*v[2] + u[3]*v[3]
Given the vectors u=[[5],[3],[-4]] and v=[[1],[0],[-5]], we can find the Euclidean inner product by substituting the values into the formula:
Euclidean inner product = (5)*(1) + (3)*(0) + (-4)*(-5)
= 5 + 0 + 20
= 25
Therefore, the Euclidean inner product of the given vectors is 25.
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Prove that these two statements have the same slope: y = -3x - 8 and 3x + y = -8
Answer:
When you make 3x + y = -8 into a slope intercept equation it will become the same question
3x + y = -8
-3x | -3x
y = -3x - 8
Sketch the region corresponding to the statement P(z 1.4) Shade:Left of a value -M Click and drag the arrows to adjust the values. -3 -2 -1 0 Sketch the region corresponding to the statement P(-c < < c) = 02. Shade: Left of a value.Click and drag the arrows to adjust the values. Sketch the region corresponding to the statement P( ckzk c) -0.2 Shade: Left of a value Click and drag the arrows to adjust the values. -3 -2 -1 0 License Points possible: 5 This is attempt 5 of 5. Score on last attempt (0, 0). Score in gradebook: (2.5, 0) Out of: (2.5, 2.5) Submit
The region corresponding to the statement P(z<1.4) is the area to the left of z=1.4 on a standard normal distribution. This represents the probability of obtaining a z-score less than 1.4.
The region corresponding to the statement P(-c < z < c) = 0.2 is the area between two values, -c and c, on a standard normal distribution that contains 20% of the total area under the curve. This represents the probability of obtaining a z-score between -c and c.
The region corresponding to the statement P(|z|>c) = 0.2 is the area to the left of z=-c and to the right of z=c on a standard normal distribution that contains 20% of the total area under the curve. This represents the probability of obtaining a z-score that is greater than c or less than -c.
The first statement, P(z < 1.4), refers to the probability that the random variable z is less than 1.4. To sketch this region, we would shade the area to the left of the value 1.4 on the number line.
The second statement, P(-c < z < c) = 0.2, refers to the probability that the random variable z is between -c and c, and that this probability is equal to 0.2. To sketch this region, we would shade the area between -c and c on the number line, and adjust the values of c until the shaded area represents 0.2 of the total area under the curve.
The third statement, P(c < z < k) = -0.2, refers to the probability that the random variable z is between c and k, and that this probability is equal to -0.2. To sketch this region, we would shade the area between c and k on the number line, and adjust the values of c and k until the shaded area represents -0.2 of the total area under the curve.
It is important to note that probabilities cannot be negative, so the third statement is not valid. The shaded area should always represent a positive value between 0 and 1.
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Composition rational Feb 19, 7:37:30 PM Find the composition g(f(x)) given that f(x)=(1)/(x+3) and g(x)=x^(2)
The composition of the two given rational functions is another rational function, [tex]g(f(x)) = 1/((x+3)^2)[/tex].
The composition of two functions, f(x) and g(x), is denoted as g(f(x)) and is defined as the function that results from applying g(x) to the output of f(x). In other words, the composition of two functions is the function that results from plugging one function into another function.
To find the composition [tex]g(f(x))[/tex], we need to substitute the expression for [tex]f(x)[/tex] into the expression for [tex]g(x)[/tex] wherever we see an "x".
Given that [tex]f(x)=(1)/(x+3)[/tex] and [tex]g(x)=x^2[/tex], the composition [tex]g(f(x))[/tex] can be found as follows:
[tex]g(f(x)) = g((1)/(x+3)) = ((1)/(x+3))^(2) = (1^(2))/((x+3)^(2)) = 1/((x+3)^(2))[/tex]
Therefore, the composition [tex]g(f(x)) = 1/((x+3)^2)[/tex].
In conclusion, the composition of the two given rational functions is another rational function, [tex]g(f(x)) = 1/((x+3)^2)[/tex].
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The restrictions on x, when (x+4)/(5x-1) is divided by (3x+12)/(6x), can be written in the form x!
The restrictions on x are x = 1/5 and x = 0.
The restrictions on x, when (x+4)/(5x-1) is divided by (3x+12)/(6x), can be found by looking at the denominators of each fraction. The restrictions are values of x that would make the denominator equal to zero, which would make the fraction undefined.
For the first fraction, (x+4)/(5x-1), the restriction is when 5x - 1 = 0. Solving for x, we get:
5x = 1
x = 1/5
For the second fraction, (3x+12)/(6x), the restriction is when 6x = 0. Solving for x, we get:
x = 0
Therefore, the restrictions on x are x = 1/5 and x = 0. These values of x cannot be used in the original expression because they would make the denominator equal to zero and the expression undefined.
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Fill in the blank so that the ordered pair is a solution of y=22−9x
.
One possible ordered pair that is a solution of the equation y = 22 - 9x is (2, 4).
What is an ordered pair, and how can we find solutions of a linear equation?
An ordered pair is a pair of numbers (x, y) that represents a point on a coordinate plane. In algebra, we often use ordered pairs to represent solutions of equations, where the x-coordinate represents a variable and the y-coordinate represents the corresponding value of the expression.
To find solutions of a linear equation, we can substitute different values of the variable into the equation and solve for the corresponding values of the expression.
Find the ordered pair:
We are given the equation y = 22 - 9x, and we want to find an ordered pair that is a solution of the equation. To do this, we can choose a value of x and then use the equation to find the corresponding value of y.
Let's choose x = 2. Then, we can substitute x = 2 into the equation and solve for y:
y = 22 - 9(2)
y = 22 - 18
y = 4
Therefore, when x = 2, y = 4, which means the ordered pair (2, 4) is a solution of the equation.
We could also check this by graphing the equation and verifying that the point (2, 4) lies on the line represented by the equation.
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The table for the quadratic functions f(x) and g(x) are given.
x f(x) g(x)
−6 36 4
−3 9 1
0 0 0
3 9 1
6 36 4
Determine the type of transformation and the value of k.
1. g(x) = 3f(x)
2. g(x) = f(3x)
3. g of x equals one third times f of x
4. g of x equals f of one third times x
The value of k is 1, which is the value of g(x) (and f(x)) when x = -3 or x = 3.
We can determine the type of transformation and the value of k for each of the aforementioned functions using the tables provided for f(x) and g(x).
g(x) = 3f(x) (x)
Here, there occurs a vertical stretch/compression transformation. The function g(x) is a three-fold vertical expansion or contraction of f(x). G(x) has a value of 4, which is identical to the value of k, whether x = -6 or = 6.
g(x) = f(3x) (3x)
Here, there occurs a horizontal stretch/compression transformation. A horizontal stretch or compression of f(x) by a factor of 1/3 results in the function g(x). When x = -3 or x = 3, the value of k is 1, which is also the value of g(x) and f(x).
g(x) = (1/3)f(x) (x)
Here, there occurs a vertical stretch/compression transformation. A vertical stretch or compression of f(x) by a factor of 1/3 results in the function g(x). G(x) has a value of 4, which is identical to the value of k, whether x = -6 or = 6.
g(x) = f(x/3)
Here, there occurs a horizontal stretch/compression transformation. The function g(x) is a three-fold horizontal stretching or compression of f(x). When x = -3 or x = 3, the value of k is 1, which is also the value of g(x) and f(x).
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Given the linear equation of y=8, comparing to the equation of
the straight line, what is c?
A. 0
B. 1
C. 8
D. Cannot be calculated
The value of c is 8 (option C).
Determine he value of cTo find the value of c in a linear equation, we can compare the given equation to the standard form of a linear equation, which is y = mx + c.
In this case, the given equation is y = 8. If we compare this to the standard form, we can see that m (the slope) is 0 and c (the y-intercept) is 8.
Therefore, the value of c is 8.
So, the correct answer is C. 8
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Let l, m, and n be three lines; if n Im and m ll then 1 || n. Let l, m, and n be three lines; if I || m and m || n then 1 || n. Let k, l, m, and n be four lines; if k 11,11 m and m In then k \ n.
This property can be applied to any number of lines as long as they are all parallel to the same line.
The statement "if n I| m and m ll then 1 || n" is not valid as the symbols used are not correct. The correct statement should be "if l || m and m || n then l || n". This means that if line l is parallel to line m and line m is parallel to line n, then line l is also parallel to line n. This is known as the transitive property of parallel lines.
Similarly, the statement "if k 11,11 m and m In then k \ n" is not valid as the symbols used are not correct. The correct statement should be "if k || l, l || m, and m || n then k || n". This means that if line k is parallel to line l, line l is parallel to line m, and line m is parallel to line n, then line k is also parallel to line n. This is also an application of the transitive property of parallel lines.
In conclusion, the transitive property of parallel lines states that if two lines are parallel to the same line, then they are also parallel to each other. This property can be applied to any number of lines as long as they are all parallel to the same line.
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Mr. and Mrs. Doran have a genetic history such that the probability that a child being born to them with a certain trait is 0.82. If they have four children, what is the probability that exactly two of their four children will have that trait? Round your answer to the nearest thousandth.
Using binomial distribution, the probability of exactly two of their four children having the trait is 0.13 (rounded to the nearest thousandth).
What is the probability that a child born to them with a certain trait is 0.82This is a binomial distribution problem with n = 4 trials (number of children) and p = 0.82 probability of success (having the trait) for each trial.
The probability of exactly two children having the trait can be calculated using the binomial distribution formula:
P(X = 2) = (4C2) * 0.82^2 * (1 - 0.82)^(4-2)
where (4C 2) is the number of ways to choose 2 children out of 4.
Using a calculator or statistical software, we get:
P(X = 2) = (4 C 2) * 0.82^2 * (1 - 0.82)^(4-2)
= 6 * 0.82^2 * 0.18^2
= 0.13
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The sum of two number is 40. The larger number is 8 more than the smaller number. What are the numbers?
Answer:
16 + 24
Step-by-step explanation:
The solids are similar. Find the missing dimension.
d
12ft.
8in.
3ft.
Answer:
32 in
Step-by-step explanation:
You want the missing diameter of the smaller of two similar cylinders, where the larger is 12 ft in diameter and 3 ft high, while the smaller is 8 inches high.
Similar figuresThe linear dimensions of similar figures have the same ratio.
The ratio of the diameter to the height of the larger figure is ...
(12 ft)/(3 ft) = 4
The smaller figure will also have a diameter that is 4 times the height:
d = 4 × 8 in = 32 in
The missing dimension is 32 inches.
Someone please help me with this I got 0 clue what I'm doing with it
Answer:
3rd choice down
f(x) = -1/2x + 8
Step-by-step explanation:
find the slope of the line using the 2 given points:
slope = m = (8-14) / (0--12) = -6/12 = -1/2
read the y intercept right off the graph at point (0,8):
b = 8
f(x) = mx + b = -1/2x + 8
Calculate the expected value of the scenario.
xi P(xi)
11 0.24
22 0.31
33 0.01
44 0.15
55 0.29
The expected value of this scenario is 4.19.
The Expected value (EV), which is based on a random variable's probability distribution, describes the long-term average level of that variable. The expected value of a stock or other investment is a crucial factor in investing and is taken into account while performing scenario analysis.
To calculate the expected value, we need to multiply each outcome xi by its respective probability P(xi), and then add up all of these products. So, we have:
Expected value = 1(0.24) + 2(0.22) + 3(0.31) + 4(0.01) + 5(0.16) + 6(0.29)
Expected value = 0.24 + 0.44 + 0.93 + 0.04 + 0.80 + 1.74
Expected value = 4.19
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HELP!!!!!!!!!!!!!!!!!!!!
The value of m and n is 5√2 and 5 respectively.
What is Pythagoras' theorem?The hypotenuse side of a right-angled triangle's square is equal to the sum of the squares of its other two sides, according to Pythagoras' Theorem.
A right-angled triangle.
A 45° angle is found in a triangle.
The triangle is therefore a right-angled isosceles triangle.
Hence, n = 5.
To find m:
Apply Pythagoras' theorem,
m = √{n² + 5²}
m = √(25 + 25)
m = √50
m = 5√2
Therefore, m = 5√2.
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the product (2x^(4)y)(3x^(5)y^(8))is equivalent towhat polynomial must be added to x^(2)-2x+6 so that the sum is 3x^(2)+7x
The polynomial that must be added to x^(2)-2x+6 is 2x^(2) + 9x - 6.
To find the product of the two polynomials (2x^(4)y)(3x^(5)y^(8)), we need to use the distributive property and combine like terms.
The distributive property states that a(b+c) = ab+ac. So, we can distribute the first polynomial to each term in the second polynomial:
(2x^(4)y)(3x^(5)y^(8)) = (2x^(4)y)(3x^(5)) + (2x^(4)y)(y^(8))
Next, we can combine like terms by adding the exponents of the variables:
= 6x^(4+5)y^(1+8)
= 6x^(9)y^(9)
So, the product of the two polynomials is 6x^(9)y^(9).
To find the polynomial that must be added to x^(2)-2x+6 so that the sum is 3x^(2)+7x, we can set up an equation:
x^(2)-2x+6 + (a+bx+cx^(2)) = 3x^(2)+7x
Then, we can rearrange the equation to solve for the polynomial:
a+bx+cx^(2) = 3x^(2)+7x - x^(2) + 2x - 6
a+bx+cx^(2) = 2x^(2) + 9x - 6
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PLS HELP I NEED THIS DONE TODAY
Answer:
The answer is going to be 4r (2r + 3)
Convert the following phrase into a mathematical expression. Use x as the variable, and combine like terms.
Eight times a number added to −7, subtracted from triple the sum of four times the number and 7
The expression is ____.
Answer:
3(4x + 7) - (8x - 7)
Step-by-step explanation:
(a)Let Q be the quotient of the ring R=Z*Z by the ideal I= 2Z*2Z.
What is Q four elements?
(b) Use the Fundamental Homomorphism Theorem to show Q is
equivalent to Z2*Z2?
By the Fundamental Homomorphism Theorem, Q is isomorphic to Z2*Z2.
a) Q is the quotient ring R/I, which has four elements: [0]_I, [1]_I, [2]_I, and [3]_I. Here, [x]_I is the equivalence class of x in R/I.
b) Using the Fundamental Homomorphism Theorem, we can show that Q is equivalent to Z2*Z2. Since I is a normal subring of R, the quotient ring Q can be written as Q = R/I. Then the homomorphism defined by φ: R → Z2*Z2, where φ(r) = (r mod 2, r mod 2) is onto and I is the kernel of φ. Thus, by the Fundamental Homomorphism Theorem, Q is isomorphic to Z2*Z2.
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