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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.
1,849 Questions
What is the answer to 5 plus b3?
5 + b3 cannot be solved since it is not an equation. It is simply a combination of 2 terms.
State the principal of duality in boolean algebra?
If {X,R} is a Partially Ordered Set, then {X,R(inverse)} is also a Partially Ordered Set.
Why algebra is known as algebra?
Algebra was invented by the Muslim mathematician Al-Khwarizmi and is the Arabic word (aljabr) for "equation".
That isnt really a normal question so let me tr to explain: in order to find the volume (v) of a rectangular prism (3d rectangle ) you multiply length of one of the sides (use either the "top" or "bottom" )by its width and then multiply that by the Hight of the rectangular prism. Hope this helps!
the number that is in the middle , it holds the equal distance from the greatest number and the least number
EX : set: [1,5,2,6,4]
then : [1,2,4,5,6]
then the median is 4 because it is 2 values away form the greatest number which is 6 and the least number 1
If you graphed this equation y=x-3 you would get a line that crosses the y-axis at -3 and slants up to the right at a slope of 1/1. There would not be anything that blocks it from going higher or lower at any point so, its range is "All Real Numbers."
What is the value of a 32 Meriden 231489?
depends on condition Guess between $40 - $300 but would have to be in great condiction. It was really just an inexpensive yet reliable pistol sold by Sears in the early 1900's.
If you are talking about a2+b2=c2 then that is Pythagorean theorem.
What are some real life applications of a pattern in algebra?
Read word problems in a math book.
Joe pays 25 dollars for 3 cds. At the same rate, what should he pay for 105 cds?
What is meaning of limit of function?
Finding a limit is the process of allowing a variable to 'approach', or get very close to some numerical value and finding what effect this has on the function.
Ex: f(x) = (x2 -4)/(x-2)
if you attempt to find the value of the function when x =2, you will be attempting to divide by zero, which is undefined.
But find the limit and as x-->2 , f(x) --> 4
you can use x = 1.9 then x = 1.99, then x = 1.999 (approaching 2)
and f(1.9) = 3.9, f(1.99)=3.99, f(1.999) = 3.999
So you can see that the function is getting closer to 4
This is a great question!
First off, we have to define what is meant by ordering numbers. It turns out that there is no possible way to prove that numbers even can be ordered, we simply must assume that they can be. Mathematically speaking, this means that the ordering of numbers must be an axiom, or an unprovable statement which is considered to be the truth.
This is the most famous axiom in modern mathematics, and also the most controversial. It is known by several different names: the axiom of choice, Zorn's lemma, Zermelo's theorem, and the well-ordering theorem, to name a few. It is also the ninth and final axiom of ZFC (see link below), which is the axiomatic set theory that we base our entire system of mathematics on.
The axiom of choice basically says that a set is defined as being well-ordered by a strict total order, if every non-empty subset of the set has a least element under the ordering. OK, there's our definition of order, but how do you actually construct a strict total order? Well, I'm going to show you how below, but it involves a little set theory. If you're unfamiliar with basic set theory, follow the corresponding link below.
First we need to make a partial ordering. Fortunately, that's already been done for the numbers that we use via the relation, "less than or equal to," symbolized as ≤. Here's how the partial ordering is made:
Let x, y, z Є N, where N is the set of natural numbers (0, 1, 2, 3, 4, ...). If N is partially ordered under the relation ≤, then the following three rules must hold.
1) x ≤ x for any x Є N
2) If x ≤ y and y ≤ x then x = y for any x, y Є N
3) If x ≤ y and y ≤ z then x ≤ z for any x, y, z Є N
Now, in order to turn this partial order into a strict total order, only one more thing is required.
If N is to be considered a strict total ordering under the relation ≤, then either x ≤ y or y ≤ x for all x, y Є N.
So, finally, is N well-ordered? Before I answer that, I'm going to give a quick example of a well-ordered set.
The set {1, 2, 3} is well-ordered under the relation ≤. Why? Well, let's look at every possible subset of {1, 2, 3}: {Ø}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, and {1, 2, 3}. For {1, 2, 3} to be well-ordered, each one of these sets must have a "least" element under the relation ≤. Well the first four sets listed are trivial since they either have 0 or 1 element in them. The next three listed all have a least element, specifically 1, 1, and 2 (1 ≤ 2 but 2 is not ≤ 1, thus 1 has a lower ordering than 2, for example). Finally, the set {1, 2, 3} has 1 as a least element. So, every subset of {1, 2, 3} has a least element, therefore, by the definition written above, {1, 2, 3} is a well-ordered set under the relation ≤.
Well, it should be obvious that N is a well-ordered set since N is basically the set {1, 2, 3} extended out to infinity (with 0 included). I'm obviously not going to list every subset of N in order to show that they all have a least element, but if you'd like to try and find one that doesn't, be my guest.
Lastly, I'll toss you a (hopefully) curiosity-inducing bone. It turns out that the ordering of N can be extended further, into the realm of group theory in fact. Besides what is written above, N is also the only naturally ordered semigroup in mathematical existence, within isomorphism of course.
What is the value of a 38 sw model 12-2?
Depends on condition, finish, box, papers, etc., value range 250-500 USD.
Is there a concept in mathematics that is impossible to prove a thing does not exist?
There is no such concept because it is not true. There are many mathematical proofs based on proving the non-existence of a thing.
The following example should demonstrate.
I want to prove that the biggest number does not exist.
Let us suppose it does exist, and let us call it B (for biggest).
But then consider B + 1.
B + 1 is bigger than B (this can be proved as well).
So B cannot be the biggest number.
That is, there is no such number B. In other words, the biggest number does not exist.
Is the input in a function table supposed to be the same as the rest of the input?
No, because then the output would be the same as the rest of the output(s).
Where does algebra derive from?
The word "algebra" derives from the work of Muhammad ibn Mūsā al-Khwārizmī, a Persian mathematician. In 830 he wrote a book entitled Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala, in which he brought together various algebraic procedures the methods. The term "al-gabr" is believed to mean something like "restoration" or "completion" and refers to the translation of subtracted items to the other side of the equation.
There is no square root of negative 8, negative numbers can never be square rooted since no two of the same number can go into a negative number, for example 16's square root is 4 and -4 since 4x4=16 and -4x-4=16, but -16 doesn't have a square root since -4 just turns into a positive when multiplied to itself.
Algebraic Properties of Matrix Operations. In this page, we give some general results about the three operations: addition, multiplication.
