Creating a Theorem with 4 Real Number Axioms and 2 Laws of Logic

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In summary, the conversation discusses four axioms in real numbers and two laws of logic. The cancellation theorem is also mentioned, which states that if x+y = x+z, then y=z. The possibility of creating a theorem is brought up, but it is not specified what that theorem may be. The speaker also asks for clarification on whether this is a homework help question or a general inquiry.
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evagelos
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Given 4 axioms in real Nos:

1) 1x = x ,for all x

2) x+0 = x ,for all x

3) (x+y)z = xz+yz ,for all x,y,z

4) xy =yx ,for all x,y

The cancellation theorem : [tex]\forall[x+y = x+z\Longrightarrow y=z][/tex]

And two laws of logic:

1) The law of Universal Elimination

2) The law of substitution.

Can we create a theorem?.

If yes ,what that theorem may be??
 
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Is this homework help or general inquiry? If homework help, what progress have you made before we answer? If simply general inquiry, I'll be more than happy to jump right in!
 
  • #3
BWElbert said:
Is this homework help or general inquiry? If homework help, what progress have you made before we answer? If simply general inquiry, I'll be more than happy to jump right in!

General inquiry.
 

1. What are the four real number axioms?

The four real number axioms are closure, associativity, commutativity, and distributivity. Closure states that the result of any operation on real numbers is also a real number. Associativity states that the grouping of numbers in an operation does not affect the result. Commutativity states that the order of numbers in an operation does not affect the result. Distributivity states that multiplication distributes over addition.

2. What are the two laws of logic?

The two laws of logic are the law of identity and the law of non-contradiction. The law of identity states that a statement is always true if it is logically equivalent to itself. The law of non-contradiction states that a statement and its negation cannot both be true at the same time.

3. How do the four real number axioms and two laws of logic relate to creating a theorem?

The four real number axioms and two laws of logic provide a foundation for creating mathematical theorems. These principles ensure that the operations and statements used in the theorem are valid and logical.

4. Can a theorem be created with fewer or more axioms and laws of logic?

Yes, a theorem can be created with fewer or more axioms and laws of logic. However, using too few axioms may not provide enough structure for the theorem to be applicable to a wide range of problems. On the other hand, using too many axioms may make the theorem overly complex and difficult to understand.

5. What are some challenges when creating a theorem using real number axioms and laws of logic?

One challenge is ensuring that the theorem is applicable to a wide range of problems and not just specific cases. Another challenge is proving the theorem using rigorous mathematical reasoning. Additionally, it can be difficult to strike a balance between using enough axioms and laws to make the theorem robust, but not so many that it becomes overly complex.

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