Formal Proofs in Maths: Establishing Equivalence

In summary, formal proofs in mathematics are a rigorous method of establishing equivalence between mathematical statements using logical rules and axioms. They are important because they provide a solid and unambiguous foundation for mathematical arguments, ensure a statement is true for all possible cases, and allow for the discovery of new theorems. Formal proofs differ from informal proofs in their use of strict rules and symbols, and commonly used symbols include logical and mathematical operators. To learn how to write formal proofs, one can study logic and mathematical foundations, practice with examples, and seek guidance from experienced mathematicians or textbooks while following the rules and conventions of formal proof writing.
  • #1
solakis1
422
0
From the book "FORMAL PROOFS IN MATHS "(Amazon.com),page 101 ,exercise19 ,Iread:

Establish the equivalence between:

\(\displaystyle 0<1\),.........\(\displaystyle 0<A\Longrightarrow 0<\frac{1}{A}\),............\(\displaystyle AC<BC\wedge 0<C\Longrightarrow A<B\)
 
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  • #2
solakis said:
From the book "FORMAL PROOFS IN MATHS "(Amazon.com),page 101 ,exercise19 ,Iread:

Establish the equivalence between:

\(\displaystyle 0<1\),.........\(\displaystyle 0<A\Longrightarrow 0<\frac{1}{A}\),............\(\displaystyle AC<BC\wedge 0<C\Longrightarrow A<B\)

Please post the solution you have ready.
 
  • #3
MarkFL said:
Please post the solution you have ready.

I am sorry but the book where i took that inequality does not give a solution, so let that be an unsolved challenge question
 

FAQ: Formal Proofs in Maths: Establishing Equivalence

What are formal proofs in mathematics?

Formal proofs in mathematics are a rigorous method of establishing equivalence between mathematical statements. They involve using logical rules and axioms to logically deduce a conclusion from a set of premises.

Why are formal proofs important in mathematics?

Formal proofs are important in mathematics because they provide a solid and unambiguous foundation for mathematical arguments. They ensure that a statement is true for all possible cases, rather than just a few examples, and they allow for the discovery of new theorems and solutions.

How are formal proofs different from informal proofs?

Formal proofs are different from informal proofs in that they follow a strict set of rules and symbols, while informal proofs may use natural language and less structured reasoning. Formal proofs also provide a more rigorous and complete justification for a statement's truth.

What are some common symbols used in formal proofs?

Common symbols used in formal proofs include logical operators such as "and" (∧), "or" (∨), "implies" (→), and "not" (¬), as well as mathematical symbols such as =, <, >, and ∃ (there exists).

How can one learn to write formal proofs?

One can learn to write formal proofs by studying logic and mathematical foundations, practicing with example problems, and seeking guidance from experienced mathematicians or textbooks. It is also important to understand and follow the rules and conventions of formal proof writing.

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