Math logic is a branch of mathematics that studies how to make arguments, states the obvious statement, and ensures that reasoning used in arguments is true. In mathematical logic, formal prove and Theorem Playing a central role. Formal evidence is a systematic process to show the truth of a mathematical statement, while theorem is a statement that's been proven correctly based on axioma and the rules of logic.

Basic Mathematical Logic

Mathematical logic built on the basic structure of proposition, that is a statement that has the value of truth (right or wrong). For example, "2 is even" is the right proposition, whereas "3 is even" is the wrong proposition.

In logic, proposition can be combined using such logic operators and (coughs), or (disconnects), if... then (implication), and No. (negation). The combination of these proposition forms a more complex statement that can be proven through formal-proof techniques.

Formal Proofing

Formal prove is a logical series of steps leading from assumptions or acsioma to a conclusion, with each step based on predetermined rules of logic. There are some types of proof that are common in mathematical logic:

1. Live proofIn direct proof, we start off assuming and gradually apply the rules of logic to reach conclusions. For example, to prove that if nnn is an even number, then even n2n ^ 2n2, we could use even definition and simple algebra to reach this conclusion.
2. Indirect proof (contraction): In this proof, we proved the implications of P QP rightarrow QP Tac Q by proving the contraction, which is # Pneg Q rightarrow neg P # ^ P. This means that if we can prove that QQQ is wrong, then PPP is also wrong, then we've proved that P QP rightarrow QP is correct.
3. Proofing with contradictionThe proof of contradiction begins by assuming that the statement we want to prove is false, and then shows that those assumptions cause contradictions. That way, we conclude that the initial statement has to be true.
4. Proving math inductionThis technique was used to prove statements involving integers or series. Proving with induction consists of two steps:
   • Base step, where we prove that the true statement for an initial value (e.g. n = 1n = 1n = 1).
   • The induction step, where we show that if the true statement for n = nkn = k, then the statement is also true for n = k + 1n = k + 1n = k + 1n = k + 1.

Theorema and Aksioma

Theorem is a mathematical statement that has proven right through formal proving. A theorem is usually stated in form implication, that if certain conditions are fulfilled, a conclusion will take effect. For example, Pythagoras Theorem It states that for each triangle my elbow, the number of squared of the two-sided length is equal to the squared of the length of its mirrored side.

Every theorem is based on axioma, which is an accepted statement as a correct without proof. Axioma acts as the basis of the entire mathematical system, and from this is another theorem that can be built through logical proof. One of the famous axioma systems is Euclid axioma in geometry, which is the basis for Euclidean's geometry.

In addition to theorem and axioma, we also know lemma and Kororan. Lemma is the result of the middleman used to help prove a larger theorem. Kororan is a direct conclusion of a proven theorem.

Famous Theorem Example in Mathematics

Some are famous theorists in mathematical logic and other mathematical fields include:

1. Incomplete Theorem GödelThis theorem states that in any acsiomatic system strong enough to handle integers, there's always a statement that can't be proven right or wrong in that system. It means that math is incomplete; there's always a limit to what is proven.
2. Pythagoras TheoremThis theorem states that in the elbow triangle, the square of the slash-length is equal to the square number of the other two sides. It's one of the most fundamental theorists in geometry.
3. Fundamental Arimetics Theorem: This theorem states that any positive integer greater than 1 can be specified as the result of times prime numbers in a unique way, except the sequence of factorization.

Mathematical Logic Application

Mathematical logic is not only important in mathematical proof, but also in Science computer, philosophy, and linguistic. In computer science, logic is used in development algorithms and programming languages. Formal prove it also becomes the basis to verify that software functions correctly according to the specified specifications.

Conclusion

Math logic is the essential foundation of all mathematical branches. Through formal prove, mathematicians can make sure that theorem and statements - statements made based on axioma and logical rules are correct. By studying mathematical logic, we're learning not only how to prove mathematical statements, but also understanding the basic reasoning applied to various areas of science.

source: Enderton, H. B. A Mathematical Introduction to Logic. Inducal Press.