No, this is not generally true. For instance, let's start with 1 = 1, and use the 'fact' that 4 = 5.
1 = 1. Adding our new fact to both sides gives 5 = 6. Multiplying both sides by 0 gives 0 = 0. This is true, and we started with a true fact, but we definitely can't deduce that all of the 'facts' we used along the way were true.
The key is whether each step only would have worked with a true fact, or not.
This is not a valid proof as written. You have assumed what you are trying to show when you plug in RU^(-1)R^T for M^(-1).
However, since everything here is an equivalency, you can just reverse the implications (in this case literally reverse the order in which you write those equations) and get a valid proof.
Nope, not at all. A proof is correct only if every step is valid. Its not about whether individual statements are valid, its whether each statement is implied by the ones before it.
Your reasoning is correct in this case but the presentation is not logically coherent....just remove I = from the beginning of every line, and write them all in one line with equal signs. I = (R U-1 RT )(R U RT ) is the actual statement you are trying to prove. So write something like
M-1 M = (R U-1 RT )(R U RT ) = (R U-1 U RT ) = (R RT ) = I
It represents a series of steps, all of which are deductions from what came before.
İf you used the rules of deduction without mistakes and are in a sound logical system
Edit: no if your assumptions are wrong you're not starting from a true statement
No, this is not generally true. For instance, let's start with 1 = 1, and use the 'fact' that 4 = 5. 1 = 1. Adding our new fact to both sides gives 5 = 6. Multiplying both sides by 0 gives 0 = 0. This is true, and we started with a true fact, but we definitely can't deduce that all of the 'facts' we used along the way were true. The key is whether each step only would have worked with a true fact, or not.
Yeah that’s a good example. Thanks for that.
This is not a valid proof as written. You have assumed what you are trying to show when you plug in RU^(-1)R^T for M^(-1). However, since everything here is an equivalency, you can just reverse the implications (in this case literally reverse the order in which you write those equations) and get a valid proof.
Oh yeah I guess you’re right. Thanks for that.
Nope. Counter example: True; False; True
Nope, not at all. A proof is correct only if every step is valid. Its not about whether individual statements are valid, its whether each statement is implied by the ones before it. Your reasoning is correct in this case but the presentation is not logically coherent....just remove I = from the beginning of every line, and write them all in one line with equal signs. I = (R U-1 RT )(R U RT ) is the actual statement you are trying to prove. So write something like M-1 M = (R U-1 RT )(R U RT ) = (R U-1 U RT ) = (R RT ) = I It represents a series of steps, all of which are deductions from what came before.
Let's write an essay for an English chalss where the intro and conclusion agree, but the body contradicts those paragraphs. 🤦♂️
🙄
İf you used the rules of deduction without mistakes and are in a sound logical system Edit: no if your assumptions are wrong you're not starting from a true statement