Proof for a higher level math class

In summary, the statement "ac | bc if and only if a | b" can be proven using examples, but not to prove a statement. To prove the direction of ac | bc ==> a | b, it can be shown that if ac divides bc, then b must also be an integer multiple of a.
  • #1
aesailor
6
0

Homework Statement



ac | bc if and only if a | b (Note that this is really two implications.)



2. The attempt at a solution

The only way I can see going about this proof is by using examples. I know that in ac | bc the c will cancel out in both, but I don't know how to properly word the proof. Any suggestions would be greatly helpful.
 
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  • #2
aesailor said:

Homework Statement



ac | bc if and only if a | b (Note that this is really two implications.)



2. The attempt at a solution

The only way I can see going about this proof is by using examples. I know that in ac | bc the c will cancel out in both, but I don't know how to properly word the proof. Any suggestions would be greatly helpful.
You can use examples to disprove a statement (these are called counterexamples), but you can't use examples to prove a statement.

For the direction ac | bc ==> a | b, since ac divides bc, then bc must be some integer multiple of ac, so bc = k * ac, for some integer k. Can you show that this implies that b must be an integer multiple of a?
 

Related to Proof for a higher level math class

1. What is the purpose of proof in higher level math classes?

The purpose of proof in higher level math classes is to provide rigorous and logical evidence to support mathematical statements and theorems. This allows for a deeper understanding of mathematical concepts and helps to build a strong foundation for future mathematical studies.

2. What are some common methods used to prove mathematical statements?

Some common methods used to prove mathematical statements include direct proof, proof by contradiction, and proof by induction. Direct proof involves using logical steps to show that a statement is true. Proof by contradiction involves assuming the opposite of a statement and showing that this leads to a contradiction. Proof by induction is a technique used to prove statements that follow a specific pattern.

3. How can one improve their skills in writing proofs?

One can improve their skills in writing proofs by practicing regularly and seeking feedback from instructors or peers. It is also helpful to carefully study and understand the structure and logic behind well-written proofs in textbooks and other resources.

4. Can proofs in higher level math classes be challenging for students?

Yes, proofs in higher level math classes can be challenging for students as they require a strong grasp of mathematical concepts and logical reasoning skills. However, with practice and dedication, students can improve their ability to write and understand proofs.

5. Are there any tips for approaching proof-based problems in exams?

One tip for approaching proof-based problems in exams is to carefully read and understand the problem statement and any given information. It can also be helpful to create a plan or outline for the proof before starting to write. Additionally, breaking down the proof into smaller, manageable steps can make it easier to follow and identify any errors.

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