# Abstract Algebra

## Have you ever wondered what would happen if you redefined some of the basic rules of algebra so that concepts you took for granted, like AB = BA, no longer apply? Abstract algebra does just that through the study of the properties that define algebraic structures. Post all questions about fields, rings, group theory, vector spaces, and the axioms that define them into this category.

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Abstract Algebra

# What is the significance of Boolean algebra?

It is used in science and proofs of proofs in maths, taking an example:

Hypothesis: Bananas are red.

(There exists) x x (is an element of) Red (and) Banana

Banana (implies) Red

(not) Red (implies) (not) Banana

If a yellow banana is found,

(There exists) y y(is an element of)Banana (therefore) y(is an element of) Red

There is a contradiction, as the banana must be red, but it is yellow.

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# What does symbolic mean in math?

In math, symbolic logic is simply expressing a mathematically logical statement through the use of symbols. For instance, one could always write down the phrase, "one plus one equals two," but using symbolic logic, that statement can be expressed much more succinctly as 1 + 1 = 2.

A better example is:

The indefinite integral of one divided by the quantity one minus the square of x with respect to x is equal to one half multiplied by the natural logarithm of the quotient of the quantities one plus x and one minus x with the constant of integration added to this result

Symbolically written, that statement is expressed as:

∫ [1/(1 - x2)] dx = ½ ln[(1 + x)/(1 - x)] + C,

which is a whole heck of a lot easier to write!

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# What are the types of axioms?

There are two types of mathematical axioms: logical and non-logical.

Logical axioms are the "self-evident," unprovable, mathematical statements which are held to be universally true across all disciplines of math. The axiomatic system known as ZFC has great examples of logical axioms. I added a related link about ZFC if you'd like to learn more.

Non-logical axioms, on the other hand, are the axioms that are specific to a particular branch of mathematics, like arithmetic, propositional calculus, and group theory. I added links to those as well.

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# What is contradiction in algebra?

There are many ways of interpreting "contradiction" in mathematics. Some meanings are:

• Contradiction as in proof. You attempt to give the counter-proof of the theorem, but the counter-proof fails to work.
• Contradiction as in mathematical logic. If biconditional fails, we include the slash through the double arrows pointing left and right at opposite directions.
• Contradiction as in negation of the clause.
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# Is using AssemblyLoad a static reference or dynamic reference?

Dynamic references are constructed on the fly as a result of calling various methods, such as System.Reflection.Assembly.Load. Source: .NET Framework Developer's Guide How the Runtime Locates Assemblies

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# What is the definition of payroll?

The payroll is the amount of money a company/business/establishment pays its employees at any given time -- sometimes every two weeks, or sometimes once every month. For more information, see the Related Link.

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# What is the Cartesian product of two sets?

If S and T are any two sets, then their Cartesian product, written S X T, is the set of all of the ordered pairs {s, t} such that s Є Sand t Є T.

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# What does the term abelian mean?

The term abelian is most commonly encountered in group theory, where it refers to a specific type of group known as an abelian group. An abelian group, simply put, is a commutative group, meaning that when the group operation is applied to two elements of the group, the order of the elements doesn't matter.

For example:

Let G be a group with multiplication * or addition +. If, for any two elements a, b Ð„ G, a*b = b*a or a + b = b + a, then we call the group abelian.

There are other uses of the term abelian in other fields of math, and most of the time, the idea of commutativity is involved.

The term is named after the mathematician, Niels Abel.

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# Is the set of all multiples of a positive integer n closed under multiplication?

Yes, it is.

Consider any two elements of this set, both are multiples of n, so they can be written as pn and qn for some integers p and q. Multiplying them together, we obtain pqn^2, which can be factored into (pqn)n. This result is clearly a multiple of n.

Since the product of any two multiples of n is also a multiple of n, the set of all multiples of n is closed under multiplication.

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# What are some real life applications of abstract algebra?

By definition, none. For if it did have real life applications it would not be called abstract!

+++

You could consider it as establishing the rules by which the algebra needed to perform real-life tasks works.

For a simple example, if C = AB then you could say, "Fine, suppose we call A and B the sides of rectangle than C is its area, OR if we call A and B speed and time respectively then C is the distance travelled.

In both cases the algebraic rules are the same: multiply two values and you obtain their product; but many practical applications are in fact simple products so there we have the underlying pure algebra for solving them. And from that we can use pure algebra rules to determine A or B from the others.

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# What are the LACSAP fractions?

The LACSAP fractions are a set of fractions set in a geometric pattern that are part of one of the two portfolio any International Baccalaureate - Diploma student must complete.

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# Is an invertible idempotent matrix the identity matrix?

The assertion is true.

Let A be an idempotent matrix. Then we have A.A=A. Since A is invertible, multiplying A-1 to both sides of the equality, we get A = I.

Q. E. D

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# What are some 16 Filipino values?

Some Filipino Values:

• hospitality
• industriousness or industry
• religiosity
• close family ties

utang na loob

pakikisama

close family ties

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# Is the set of all 2x2 invertible matrices a subspace of all 2x2 matrices?

I assume since you're asking if 2x2 invertible matrices are a "subspace" that you are considering the set of all 2x2 matrices as a vector space (which it certainly is).

In order for the set of 2x2 invertible matrices to be a subspace of the set of all 2x2 matrices, it must be closed under addition and scalar multiplication.

A 2x2 matrix is invertible if and only if its determinant is nonzero. When multiplied by a scalar (let's call it c), the determinant of a 2x2 matrix will be multiplied by c^2 since the determinant is linear in each row (two rows -> two factors of c). If the determinant was nonzero to begin with c^2 times the determinant will be nonzero, so an invertible matrix multiplied by a scalar will remain invertible. Therefore the set of all 2x2 invertible matrices is closed under scalar multiplication.

However, this set is not closed under addition. Consider the matrices {[1 0], [0 1]} and {[-1 0], [0 -1]}. Both are invertible (in this case, they are both their own inverses). However, their sum is {[0 0], [0 0]}, which is not invertible because its determinant is 0.

In conclusion, the set of invertible 2x2 matrices is not a subspace of the set of all 2x2 matrices because it is not closed under addition.

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# What is an example of a Tarski monster group?

I am curious as to why someone would pose a question about an advanced topic like this on this site, when a quick search for "Tarski monster group" brought up this:

http://en.wikipedia.org/wiki/Tarski_monster_group

To chase up the reference there you need access to a good college or university library.

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# What is the value of 4?

4 itself is a value.. a numeric value.......just kidding. The value of 4 is actually 6, because when you do the calculus, 4/4+85.48736=4 so thus the four is for 4.

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# How do you prove Cayley's theorem which states that every group is isomorphic to a permutation group?

Cayley's theorem:Let (G,\$) be a group. For each g Ð„ G, let Jg be a permutation of G such that

Jg(x) = g\$x

J, then, is a function from g to Jg, J: g --> Jg and is an isomorphism from (G,\$) onto a permutation group on G.

Proof:We already know, from another established theorem that I'm not going to prove here, that an element invertible for an associative composition is cancellable for that composition, therefore Jg is a permutation of G. Given another permutation, Jh = Jg, then h = h\$x = Jh(x) = Jg(x) = g\$x = g, meaning J is injective.

Now for the fun part!

For every x Ð„ G, a composition of two permutations is as follows:

(Jg â—‹ Jh)(x) = Jg(Jh(x)) = Jg(h\$x) = g\$(h\$x) = (g\$h)\$x = Jg\$h(x)

Therefore Jg â—‹ Jh = Jg\$h(x) for all g, h Ð„ G

That means that the set Ð‚ = {Jg: g Ð„ G} is a stable subset of the permutation subset of G, written as Ð–G, and J is an isomorphism from G onto Ð‚. Consequently, Ð‚ is a group and therefore is a permutation group on G.

Q.E.D.

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# What is the numbers of groups of order 8 upto isomorphisms?

There are five groups of order 8: three of them are Abelian and the other two are not.

These are

1. C8, the group generated by a where a8 = 1

2. C4xC2, the group generated by a and b where a4 = b2 = 1

3. C2xC2xC2, the group generated by a, b and c where a2 = b2 = c2= 1

4. the dihedral group

5. the quaternion group

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# A precise measurement is one that?

Is as exact as possible

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# What is axa in algebra?

a squared or a2

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# What is table algebra?

Table algebra is the branch of mathematics like geometry and others.

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# What is altitude?

== == Science Major: Altitude is height above mean sea level (MSL) or true altitude, which means in reference to what sea level is on the ground, or height above ground level (AGL), which is also called absolute altitude, which means height above the ground with no reference to the ground's height above sea level. In example, if the sea level in a certain place is 100 feet and you are 250 feet above the ground, your altitude above sea level would be 350 feet, and your altitude above ground level would be 250 feet.

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# What is the answer to page 2.11 in Punchline Algebra book A Why Did Time Seem To Go Quickly at the Glue Factory?

it was fast paste

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# What is the proof of the ''Fundamental Theorem of Algebra''?

The Fundamental Theorem of Algebra:

If P(z) = Σ­­nk=0 akzk where ak Є C, n ≥ 1, and an ≠ 0, then P(z0) = 0 for some z0 Є C. Descriptively, this says that any nonconstant polynomial over the complex number space, C, can be written as a product of linear factors.

Proof:

First off, we need to apply the Heine-Borel theorem to C. The Heine-Borel theorem states that if S is a closed and bounded set in an m-dimensional Euclidean space (written as Rm), then S is compact.

From above, P(z) = Σ­­nk=0 akzk where ak Є C, n ≥ 1, and an ≠ 0. Let m = inf{|P(z)| : z Є C} where inf is the infinum, or the greatest lower bound of the set.

From the triangle inequality, |P(reit)| ≥ rn(|an| - r-1|an-1| - … - r-n|a0|),

so limr --> ∞ |P(reit)| = ∞. Therefore there is a real number R that |P(reit)| > m + 1 whenever r > R.

If S = {reit : r ≤ R}, then S is compact in C, by the Heine-Borel Theorem; and let m = inf{|P(z)| : z Є S}. |P| is a continuous and real-valued function in S, so, using the result from another proof not done here, it has a minimum value on S; i.e., there is a value for z0 Є S that makes |P(z0)| = m. So, if m = 0 then the theorem is proved.

We're going to show that m = 0 by proving that m can't equal anything else, and since we know m exists, it has no choice but to be zero. So, suppose m ≠ 0 and let Q(z) = P(z + z0)/P(z0), z Є C.

Q is therefore a polynomial with degree n and |Q(z)| ≥ 1 for all z Є C.

Q(0) = 1 so Q(z) can be expressed via P's series as:

Q(z) = 1 + bkzk + … + bnzn where k is the smallest positive integer ≤ n such that bk ≠ 0.

Since |-|bk|/bk| = 1, there exists a t0 Є [0, 2π/k] such that eikt0 = -|bk|/bk.

Then Q(reit0) = 1 + bkrkeikt0 + bk+1rk+1ei(k+1)t0 + … + bnrneint0

= 1 - rk|bk| + bk+1rk+1ei(k+1)t0 + … + bnrneint0.

So, if rk|bk| < 1 then |Q(reit0)| ≤ 1 - rk(|bk| - r|bk+1| - … - rn-k|bn|).

That means that if we pick a small enough r, we can make |Q(reit0)| ≤ 1 which contradicts the statement above that |Q(z)| ≥ 1 for all z Є C. Therefore m ≠ 0 doesn't hold and P(z0) = 0

Q.E.D.

Another proofSuppose P has no zeroes. Then we can define the function f(z) = 1 / P(z), and f is analytic. By the proof above, P(z) tends to infinity as z tends to infinity; hence f(z) tends to 0 as z tends to infinity. So there is a disc S such that f, restricted to the outside of S, is bounded. Also by the proof above, f is bounded inside the disc as well; therefore f is bounded. Now we apply a theorem called Liouville's Theorem, which says that any analytic function which is defined on all of C and is bounded must be a constant. So f is a constant; therefore P is constant. But we were assuming that P is not constant, so this is a contradiction.

(To prove Liouville's Theorem: Suppose M is a bound for the function f, i.e. |f(z)| < M for all z. Suppose a and b are complex numbers, and we want to show f(a) = f(b). Use the theorem that f(a) = integral of f(z)/(z-a) / (2 * pi * i) around the circle of radius R and centre 0. Then, if R is sufficiently large:

|f(b) - f(a)|

= | integral, around circle, of (f(z) * (1/(z-b) - 1/(z-a))) | / (2*pi)

= | integral around circle of (f(z) * (b-a) / ((z-a)(z-b)) ) | / (2*pi)

<= (M * |b-a| / ((R-|a|)(R-|b|)) ) * (2*pi*R) / (2*pi)

The last line uses the formula |integral| <= |pathlength| * |maximum value|. Then we get |f(b) - f(a)| <= M * |b-a| * R / ((R-|a|)(R-|b|)). Letting R tend to infinity, we can prove that |f(b)-f(a)| is as small as we like; therefore f(a) = f(b).

)

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# What is a vector in Java?

Vector is a type of collection object that Java has. Vector is a class that implements the AbstractList class and is often used as an alternative to arrays since it automatically extends the length of the list unlike arrays. A Vector can contain a collection of objects of any type. But it has fallen out of use due to the rise of the more convenient ArrayList class, but Vectors are still used for their security in multi threaded environment. Vectors are thread safe but array lists are not. An important fact to note is that Stack extends the Vector class.

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