Mastering Mathematical Proofs: Solving the Universe's Integers with a-3b and a+b

In summary: But my original statement still stands, in that if a number is even then it can be written as 2k where k is an integer. I'm sorry if I made any more confusion.In summary, if a number is even, it can be written as 2k where k is an integer.
  • #1
CollectiveRocker
137
0
We are given the following statement: The universe is all integers. If a-3b is even, then a+b is even. I began off with saying that 3 cases exist: a is odd and b is even, a is even and b is odd, both a and b are odd, and both a and b are even. After this point I got really confused and lost. Can someone please point me in the right direction?
 
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  • #2
hint, if a number is even, then it can be written as [tex] 2k [/tex] where k is an integer.

since they tell you that a - 3b is even, then that means

[tex] a - 3b = 2k [/tex]

for some integer k.
 
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  • #3
think about the difference between those two expressions
 
  • #4
I don't understand what you mean when you say a-3b=2k. How does that help you?
 
  • #5
try to manipulate that equation, so you have a + b on one side, and on the other you have 2* some integer.



edit: I edited the above post, to try to make it more clear
 
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  • #6
I realize that I just sound really stupid right now; but it's been a while since I've really done any math per say. I don't think any way exists to end up with a+b on one side with 2* something on the other side. The problem exists because of the 3, no matter what happens, there is no way to separate the 3 from the b, and still end up with a+b. Is there?
 
  • #7
start by trying to obtain a + b on the left side. Remeber, anything you do on the left side you also have to do on the right side.

you can do this by adding 4b to both sides, and simplifying :smile:
 
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  • #8
Why worry about a-3b=2k? Why not just remember that the sum of two even numbers is an even number?

Doug
 
  • #9
Do you mean a+b = 2(k+2b)?
 
  • #10
Mathechyst said:
Why worry about a-3b=2k? Why not just remember that the sum of two even numbers is an even number?

Doug
where does it say that a and b are both even?
 
  • #11
CollectiveRocker said:
Do you mean a+b = 2(k+2b)?
yep... since k + 2b is an integer, then 2(k + 2b) is even, therefore, a + b is even. The trickiest part of this proof, is just knowing that if a number is even then it can be written as 2 * an integer.
 
  • #12
It doesn't. It just says that if a-3b is even, then a+b is even. Then we are supposed to prove that statement any way possible which makes logical sense.
 
  • #13
MathStudent said:
where does it say that a and b are both even?

It doesn't but (a-3b)+c is even if a-3b and c are both even.

Doug
 
  • #14
Thank you so much guys, for sticking with me.
 
  • #15
Mathechyst said:
Why worry about a-3b=2k? Why not just remember that the sum of two even numbers is an even number?

Doug

Yes. This is what I was thinking too:

a+b = (a-3b) + (4b)

Since a-3b is given as even, and since 4b is even... and since the sum of two even numbers is even we know that a+b is even.
 
  • #16
Mathechyst said:
It doesn't but (a-3b)+c is even if a-3b and c are both even.

Doug
I see what you mean ... doesn't seem like either way is faster, but its good to know both :smile: .


edit: Actually I take that back, your method is a little faster, in that you don't have any manipulating to do.
 
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1. What is the purpose of "Mastering Mathematical Proofs"?

The purpose of "Mastering Mathematical Proofs" is to teach readers how to solve mathematical proofs using the techniques of a-3b and a+b. It aims to help readers understand the underlying principles and logic behind mathematical proofs, and how they can be applied to solve complex problems.

2. How does "Mastering Mathematical Proofs" relate to the universe's integers?

"Mastering Mathematical Proofs" uses the concept of a-3b and a+b to solve problems related to the universe's integers. These techniques can be applied to various mathematical concepts, including number theory, algebra, and geometry, to better understand the fundamental properties of numbers and their relations within the universe.

3. Can "Mastering Mathematical Proofs" be used for any level of mathematics?

Yes, "Mastering Mathematical Proofs" can be used for any level of mathematics, from high school to advanced college courses. The techniques taught in this book are applicable to a wide range of mathematical concepts and can be used to solve problems at any level.

4. Is prior knowledge of mathematical proofs necessary to understand "Mastering Mathematical Proofs"?

Some prior knowledge of mathematical proofs is helpful, but not necessary, to understand "Mastering Mathematical Proofs". The book provides a comprehensive explanation of the principles and techniques used in proofs, making it accessible to readers with varying levels of mathematical knowledge.

5. Can "Mastering Mathematical Proofs" be used for self-study or is it better suited for a classroom setting?

"Mastering Mathematical Proofs" can be used for both self-study and in a classroom setting. The book includes numerous examples and practice problems to help readers apply the techniques learned. It can also be used as a supplemental text for a classroom course on mathematical proofs.

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