Basic Group Theory Proof. Looks easy, might not be.

In summary: Homework Statement Find the volume of the solid generated by revolving the region bounded by the graph x = y^2 and the lines x = 0 and y = 1 about the x-axisHomework Equations Volume of solid of revolution: V = π∫(upper limit)^2(lower limit)^2 (radius)^2 dxThe Attempt at a SolutionIn summary, the volume of the solid generated by revolving the region bounded by the graph x = y^2 and the lines x = 0 and y = 1 about the x-axis can be found by using the formula for volume of a solid of revolution and substituting the appropriate values for the upper and lower limits and the radius.
  • #1
U.Renko
57
1

Homework Statement


Let [itex]a,b[/itex] be elements of a group [itex] G[/itex]. Show that the equation [itex]ax=b [/itex] has unique solution.

Homework Equations



none really

The Attempt at a Solution



[itex]ax = b [/itex]. Multiply both sides by [itex] a^{-1}[/itex]. (left multiplication). [itex] a[/itex] is guaranteed to have an inverse since it is an element of a group.
Then [itex] a^{-1}ax = a^{-1}b[/itex] and therefore the equattion has solution [itex]x=a^{-1}b[/itex].
Since in a group, every element has an unique inverse element, it follows that the solution is unique.
I don't know, it just looks too obvious, I may be missing something:
(also: I'm not a math major. I like doing proofs just for fun, and I don't really have that much of practice (yet), so forgive any lack of rigor or something like that. )
Is that it?
 
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  • #2
U.Renko said:

Homework Statement


Let [itex]a,b[/itex] be elements of a group [itex] G[/itex]. Show that the equation [itex]ax=b [/itex] has unique solution.

Homework Equations



none really

The Attempt at a Solution



[itex]ax = b [/itex]. Multiply both sides by [itex] a^{-1}[/itex]. (left multiplication). [itex] a[/itex] is guaranteed to have an inverse since it is an element of a group.
Then [itex] a^{-1}ax = a^{-1}b[/itex] and therefore the equattion has solution [itex]x=a^{-1}b[/itex].
Since in a group, every element has an unique inverse element, it follows that the solution is unique.



I don't know, it just looks too obvious, I may be missing something:
(also: I'm not a math major. I like doing proofs just for fun, and I don't really have that much of practice (yet), so forgive any lack of rigor or something like that. )
Is that it?

That's it. It is easy.
 
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Likes 1 person
  • #3
Oh cool.
That is nice!

thanks!
 
  • #4
These "fundamental" proofs are often short and "easy". If you are just starting to do proofs, the hard part may be that the result is so "obvious" you don't understand what needs to be proved!

But these results do need proving, because often they are what make different mathematical structures have different properties. Thinking up examples of situations where a result like this is NOT true can help you understand what is special about a "group". For example, the theorem is not true for multiplication of real numbers. If a = 0, there are no solutions or an infinite number of solutions, depending on whether b is 0 or not. So the integers under multiplication are not a group.

If a,x and b are 2x2 matrices, you can find examples where there are multiple solutions when a and b are both non-zero. So whatever sort of mathematical animal 2x2 matrices are, it's not the same sort of animal as real numbers, and neither of them are groups.
 
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Related to Basic Group Theory Proof. Looks easy, might not be.

1. What is basic group theory proof?

Basic group theory proof is a mathematical technique used to prove the properties and relationships of objects within a group. A group is a set of elements that have a defined operation and satisfy certain axioms, such as closure, associativity, identity, and invertibility.

2. Why is group theory important?

Group theory is an important area of mathematics that has applications in various fields, such as physics, chemistry, computer science, and cryptography. It helps us understand the underlying structure and patterns of objects and their interactions within a group.

3. What are some common techniques used in basic group theory proof?

Some common techniques used in basic group theory proof include direct proof, proof by contradiction, proof by induction, and proof by exhaustion. These techniques involve logical reasoning and manipulation of equations to show that a statement is true for all elements in a group.

4. Can anyone learn basic group theory proof?

Yes, anyone can learn basic group theory proof with dedication and practice. It is a fundamental concept in mathematics and can be understood by following a step-by-step approach and practicing with examples and exercises.

5. What are some common misconceptions about basic group theory proof?

One common misconception is that group theory is only applicable to abstract or theoretical concepts. In reality, it has practical applications in various fields. Another misconception is that group theory is only for advanced mathematicians. However, understanding the basic principles and techniques can be helpful in solving problems in other areas of mathematics.

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