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evagelos
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Give a rigorous and then a formal proof of the theorem :
[tex]\forall x [ (-1).x = -x ][/tex]
[tex]\forall x [ (-1).x = -x ][/tex]
mXSCNT said:Hint: (-1).x = -x means that (-1).x + x = 0
This should be in a homework forum
HallsofIvy said:Once again, we return to the question, "What do you mean by 'formal proof'?"
tgt said:I would mean the kind that most people (at least all the logicians) would regard as formal.
A formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language) each of which is an axiom or follows from the preceding sentences in the sequence by a rule of inference. The last sentence in the sequence is a theorem of a formal system.
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HallsofIvy said:I think you are just talking about the kind of proof you would find in a math paper or calculus book- far from what, say, logicians would mean by "formal proof". A true "formal proof" of such a thing would probably require an entire book! How many pages did Russel and Whitehead require to prove "1+ 1= 2"?
HallsofIvy said:Another question: what algebraic system are you working in? The proof for an abstract ring or integral domain would be quite different than for the real numbers.
MXSCNT's point is that the only good reason for doing such "fiddly" stuff is practise: homework.
HallsofIvy said:How many pages did Russel and Whitehead require to prove "1+ 1= 2"?
A rigorous and formal proof is a logical and systematic way of demonstrating the truth or validity of a statement or theorem. It involves using established mathematical principles and rules of logic to show that a statement is true in all cases.
(-1).x is a mathematical expression that represents the product of -1 and x. In other words, it is the result of multiplying -1 by a number or variable x.
It is important to prove that (-1).x = -x because it is a fundamental property of multiplication and serves as the basis for many other mathematical concepts. Additionally, being able to prove this statement allows us to confidently use it in other mathematical proofs and calculations.
The proof of (-1).x = -x is typically presented using the laws and properties of algebra. It often involves manipulating and rearranging equations to show that the two expressions are equal.
Yes, here is an example of a rigorous and formal proof of (-1).x = -x:
Proof:
(-1).x = (-1) * x [Using the definition of multiplication]
= -(1 * x) [Using the distributive property]
= -x [Using the identity property of multiplication]
Therefore, (-1).x = -x, as required.