How many elements of order 2 are there in S5 and in S6? How many elements of order 2 are there in Sn? (abstract algebra)

Describe the inverse properties of real numbers and provide examples. What is useful about the inverse properties of real numbers?
Start with definitions for identity elements and then define additive and multiplicative inverses. Discuss why 0 and 1 are important in this context. Tell how and why we use inverses.
What is an example of an operation and its "additive inverse" that you use in everyday life? ( for example walking one block north and then one block south, to get back to where you started)
What is an example of an operation and its multiplicative inverse that you use in everyday life?( for example to convert from feet to inches multiply by 12. To convert inches to feet: divide by 12 or multiply by 1/ 12)

1. Use strong induction to show that every positive integer, n, can be written as a sum of powers of two: 2⁰, 2¹, 2², 2³, .....

Solve the following problems using the concepts and notation of set theory and vector spaces:

1. What is a Cartesian vector?
2. What is a real vector space?
3. Does the rational numbers fulfill the definition of a field?
4. Define linearly dependency for any set of vectors.
5. Assess the linearly dependency (or not) of the following vectors: u=<2,-1,1>, v=<3,-4,-2>, and w=<5,-10,-8>
6. Define basis

1. I am trying to determine the level of measurement of my data type?

I am looking for advice on Nominal, Ordinal, Interval, and Ratio

2. Does the data set have any categorical variables?

I am trying to Describe the data set below in very general terms?

This data consist of 8 variables: Which are

GRE Scores,

TOEFL Scores,

University Rating,

Statement of Purpose,

Letter of Recommendation Strength,

Undergraduate GPA, .

Research Experience, and

Chance of Admit.

6.14 Let f = {(x, y) ∈ R² : y = x⁵ + 4x³ + x + 1}.

1. Prove that (a) f is onto. (b) f is 1-1.

2. Prove that g = {(x, y) ∈ R² : x = y⁵ + 4y³ + y + 1} is a function. (You will need to use calculus to prove part (1).)

If R is a division ring, show that cent R forms a field.

Find the general power series solution for the differential equation
2x^2y''-xy'+(x^2 +1) y =0 about x=0

(Answer should be given to the x^4+r term)

A tank initially contains 10 liters of clean water (with no salt in it). Every minute, a solution containing one liter of water and one gram of salt is added to the tank, and two liters of fluid are drained from the tank. The fluid in the tank is mixed continuously so that the salt is distributed evenly throughout the water in the tank. How many minutes does it take for the tank to contain 2 grams of salt?

Prove or disprove: In a hyperbolic geometry, if any right triangle has its hypoteneuse and one leg congruent (respectively) to the hypoteneuse and one leg of another right triangle, then the two right triangles are congruent.

Find and solve a recurrence relation for the number of ways to stack n poker

chips using red, white and blue chips such that no two red chips are together.

Use your solution to compute the number of ways to stack 15 poker chips.

We say that an infinite sequence a0,a1,a2,a3,… of real numbers has the limit L if for every strictly positive number ε, there is a natural number n such that all the elements an,an+1,an+2,… are within distance ε of the value L. In this case, we write lim a = L.

1. Express the condition that lim a = L as a formula of predicate logic. Your formula may use typical mathematical functions like + and absolute value and mathematical relations like > or ∈ . Check that your formula has exactly two free variables: a and L.
   

2. For any two infinite sequences of real numbers a = a0,a1,a2,… and b = b0,b1,b2,… we define their sum (written a+b) to be an infinite sequence c0,c1,c2,c3,… such that ci = ai + bi for every i≥0. Prove (in complete sentences of “mathematical English”) that if we have two infinite sequences with lim a = L and lim b = M then lim (a+b) = (L+M).
   

3. Identify 5 places in your Part B proof where you took a step corresponding to a natural deduction rule, explicitly or implicitly. Each place should correspond to a different natural deduction rule.
   

4. Your proof almost certainly used well-known mathematical “facts” about natural numbers or real numbers. These “facts” are just small theorems that are so obvious or well-known that they can be used in proofs without comment. Give predicate logic formulas to specify three theorems you used. (no proofs for these, but they should all be true formulas about numbers, with no free variables.)

given the initial simplex tableau (Matrix):

x              y              s1            s2            p
6              9              1              0              0              300
5              4              0              1              0              180
-3            -4            0              0              1              0

Show the matrices produced by each pivot

Use Newton's method to find all solutions of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. (Enter your answers as a comma-separated list.)

−2x⁷ − 4x⁴ + 8x³ + 6 = 0

for unit quaternions used to represent rotations for 3D prove that :

a) quaternion conjugate corresponds to rotation inverse ( Approach 1: negate the angle in the formula for converting from the axis-angle form and apply trigonometric identities. Approach 2: convincingly discusses what negating and axis description does in terms of rotation effects and in terms of the formula)

b) negated quaternions represent the same rotation ( hint: add 2PI to the rotation angle in the axis angle formula, explain what that means in terms of orientation, and apply trigonometric identities