Proving equations are equivalent

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In summary: Assuming that the premise is true, you would need to show that the conclusion follows from the premise. In summary, the student is trying to figure out how to do a proof, but is having difficulty understanding what "by proofs" means. The student has figured out that using truth tables is a way of proving something is valid, but isn't sure how to do it.
  • #1
Gamecockgirl
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I need help proving that (A then B) then C and (A and B) then C are equivalent. Can anyone help?
 
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  • #2
There are several ways to prove such a thing. The easiest is using truth tables.
First of all, do you understand intuitively why it is true?
 
  • #3
I know why they are true and I have done the truth table however our teacher wants us to do it by proofs and I can't seem to make sense of the proof. I appreciate any guidance you can offer.
 
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  • #4
Please elaborate what "by proofs" means. Using truth table is a way of proving that expression is valid.
 
  • #5
By proof I mean using sentintial derivations where you have to use v introduction or elimination. & introduction or & elimination.
 
  • #6
So first let's consider the direct implication and work backwards. Suppose you want to prove X => Y, where X is (A => B) => C and Y is (A & B) => C. What would be your final step and which assumptions and conclusions would you need?
 
  • #7
To prove ( A then B) then C your assumption would be (A and B) then C with your conclusion being (A then B) then C. To prove (A and B) then C your assumption would be (A then B) then C and your conclusion would be (A and B) then C. But I don't know anymore than that. That is the only thing I have been able to figure out so far. I think there might be some sub proofs and some more assumptions but I don't know what.
 
  • #8
You just told me that: "to prove X, your assumption would be X with your conclusion being X. To prove Y your assumption would be Y with your conclusion being Y".

Maybe I was a bit too vague, so let me reformulate my question. You want to prove something of the form X => Y, so your final step will likely be =>I (implication introduction). So what do you start with and what do you need to prove, in order to be able to make this step?
 

1. How do you prove that two equations are equivalent?

To prove that two equations are equivalent, you must show that they have the same solution set. This can be done by simplifying both equations and showing that they are equal. Another method is to substitute values for the variables and demonstrate that the resulting expressions are equal.

2. What is the difference between equivalent equations and equal equations?

Equivalent equations have the same solution set, meaning they will produce the same result for any given values of the variables. Equal equations, on the other hand, have identical expressions on both sides and will produce the same numerical value for the same values of the variables.

3. Why is it important to prove that equations are equivalent?

Proving that equations are equivalent ensures that they will give the same result for any given set of values. This is crucial in mathematics, as it allows us to manipulate equations and solve problems more easily. Additionally, proving equivalence helps to solidify our understanding of mathematical concepts.

4. Can equations with different forms be equivalent?

Yes, equations can have different forms but still be equivalent. For example, the equations x + 2 = 5 and x = 3 have different forms, but they both have the same solution, x = 3. This is because the equation x + 2 = 5 can be simplified to x = 3, making them equivalent.

5. What are some common techniques for proving equations are equivalent?

Some common techniques for proving equations are equivalent include substitution, simplification, and using properties of equality and algebraic manipulation. These techniques involve manipulating the equations to show that they are equal or by demonstrating that they will produce the same results for any given values of the variables.

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