An Introduction to Mathematical Logic
Logic in simple words means to reason. This reasoning can be a legal opinion or even a Mathematical confirmation. Well, you can apply certain logic in Mathematics as well and solve Mathematical logic problems. Some of the basic Mathematical logical operators that you can use in your day to day life are conjunction, disjunction, and negation. You denote these Mathematical logic symbols as, ^ for representing conjunction, v for representing disjunction, and for representing negation. In this article, let us discuss some of the basic Mathematical logic, Mathematical logic formulas along with the truth table and some Math logic examples with answers.
History of Mathematical Logic
Mathematical logic arose as a subject of Mathematics in the mid-nineteenth century, combining two traditions: formal philosophical logic and Mathematics. " Mathematical logic, sometimes known as 'logistic logic, "symbolic logic,' the 'algebra of logic,' and, more recently, just 'formal logic,' is a collection of logical ideas developed in the previous (nineteenth) century using an artificial notation and a strictly deductive procedure. Prior to this, logic was learned through rhetoric, calculations, syllogism, and philosophy. The first half of the twentieth century saw a flood of important breakthroughs, accompanied by heated controversy about Mathematics' foundations.
Classification of Mathematical Logic
The Mathematical logic can be subdivided into four different fields which are as follows:
Set Theory
Recursion Theory
Model Theory
Proof Theory
Although many approaches and results are shared across various disciplines, each has its own speciality. The borders separating Mathematical logic from other branches of Mathematics, as well as the lines dividing these domains, are not always clear. Gödel's incompleteness theorem is significant not just in recursion theory and proof theory, but also in modal logic, as it led to Löb's theorem. Set theory, model theory, and recursion theory, as well as the study of intuitionistic Mathematics, use the forcing approach.
Set Theory
The study of sets, which are abstract groupings of items, is known as set theory. Cantor defined many of the fundamental concepts, such as ordinal and cardinal numbers, informally before formal axiomatizations of set theory were constructed. The first such axiomatization, credited to Zermelo, was somewhat expanded to form Zermelo–Fraenkel set theory (ZF), which is today the most extensively used Mathematical fundamental theory.
Model Theory
Model theory is the study of various formal theories' models. A theory is a collection of formulae with a specific formal logic and signature, whereas a model is a framework that provides a tangible interpretation of the theory. Model theory is closely connected to universal algebra and algebraic geometry, but its techniques emphasise logical issues rather than those of those subjects.
An elementary Class is the set of all models in a given theory; classical model theory aims to discover the features of models in a given elementary Class, or whether specific kinds of structures constitute elementary Classes.
Recursion Theory
The computability of functions from natural numbers to natural numbers is the topic of classical recursion theory. Using Turing machines, calculus, and other systems, the essential results build a resilient, canonical Class of computable functions with several independent, equivalent characterizations. The structure of the Turing degrees and the lattice of recursively enumerable sets are two further advanced conclusions.
Mathematical Logical Operators
There are three basic Mathematical logical operators that you use in Mathematics. These are:
Conjunction or (AND)
Disjunction or (OR)
Negation or (NOT)
Now, let us take a look at all these Mathematical logical operators in detail.
Mathematical Logic Formulas
Conjunction or (AND)
You can easily join two Mathematical logic statements by using the AND operand. It is also called a conjunction. You can represent it in the symbol form as ∧. In this operator, if either of the statements is false, then the result is false. If both the statements are true, then the result is true. The inputs can be two or more, but the output you get is just one.
Truth Table of the Conjunction (AND) Operator
Disjunction or (OR)
You can join two statements easily with the help of the OR operand. It is also called disjunction. You can represent it in the symbolic form as ∨. In this operator, if either of the statements is true, then the result you get is true. If both the statements are false, then the result is false. It consists of two or more inputs but only one output.
Truth Table of the Disjunction (OR) Operator
Negation or (NOT)
Negation is an operator that gives the opposite statement of the statement which is given. It is also called NOT and is denoted by ∼. It is an operation which would give the opposite result. When the input is true, the output you get is false. When the input is false, the output you get is true. It consists of one input and one output.
Truth Table of the Negation (NOT)
Mathematical Logics Problems
Now that you know about the Mathematical logic formulas, let us take a look at Math logic examples with answers.
Example 1:
Construct a truth table for the values of conjunction for the following given statements:
r: x is an odd number
s: x is a prime number
Solution:
Given:
r: x is an odd number
s: x is a prime number
Since each statement given represents an open sentence, the truth value of r∧s would depend on the value of the variable x. Since there are an infinite number of replacement values for x, you cannot list all the truth values for r∧s in the truth table. However, you can find the truth value of r∧s for the given values of x as follows:
If x = 3, r is true, and s is true. Hence, the conjunction r∧s is true.
If x = 9, r is true, and s is false. Hence, the conjunction r∧s is false.
If x = 2, r is false, and s is true. Hence, the conjunction r∧s is false.
If x = 6, r is false, and s is false. Hence, the conjunction r∧s is false.
Example 2:
Find the negation of the given statement:
Number 4 is an even number
Solution:
Consider P to be the given statement
Hence, P = 4 is an even number.
Therefore, the negation of the statement is given as
∼S = 4 is not an even number.
Hence, the negation of the statement is that 4 is not an even number.
Conclusion
Logic in simple words means to reason. This reasoning can be a legal opinion or even a Mathematical confirmation. Some of the basic Mathematical logical operators are conjunction, disjunction, and negation. In this article, let us discuss some of the basics of Mathematical logic.
FAQs on Mathematical Logic
1. What is mathematical logic?
Mathematical logic is primarily about providing a framework to communicate and explain results to each other. With the help of some commonly accepted definitions and understanding rigorously what it means when something is true, false, assumed, etc., you can explain and prove the reasons behind the things being the way they are.
2. What is the importance of mathematical logic in maths?
The goal of Mathematical logic is to improve a student's logical and analytical skills, which are required for understanding and memorising Mathematical proofs. The symbolic presentation and formal principles of Mathematical logic, on the other hand, distinguish it. For drawing conclusions, making deductions, and forming viable proofs for conjectures becoming theorems, Mathematical reasoning relies on logic and the laws of inference in logic. As a result, logic plays a crucial role in Mathematics.
3. What are the uses of mathematical logic?
The goal of Mathematical logic is to codify accurate facts and sound reasoning. Its inventors, Leibniz, Boole, and Frege, wanted to utilise it for common sense facts and reasoning, but they didn't realise that the imprecision of ideas employed in common sense language was frequently a feature, not a defect. The most significant achievement of Mathematical logic was the formalisation of Mathematical theories. The application of Mathematical logic for common sense has had little effectiveness since the common sense informatica scenario necessitates the use of inaccurate data and imprecise reasoning. Many people have given up as a result of this. Extended logical languages, as well as extended types of Mathematical logic, are gradually being conceived and developed.
4. What are the different types of mathematical logic?
The field of Mathematical logic may be split into four subfields, as follows:
Model Theory
Set Theory
Recursion Theory
Theoretical Proof
Despite the fact that many techniques and outcomes are common across disciplines, each has its own specialisation. The lines distinguishing these areas and separating Mathematical logic from other fields of Mathematics are not always apparent. Because it leads to Löb's theorem, Gödel's incompleteness theorem is important not just in recursion theory and proof theory, but also in modal logic. The forcing technique is used in set theory, model theory, and recursion theory, as well as the study of intuitionistic Mathematics.
5. What is the theory of set?
Set theory is the study of sets, which are abstract collections of elements. Before formal axiomatizations of set theory were created, Cantor defined many of the key notions informally, such as ordinal and cardinal numbers. Zermelo is credited with the first such axiomatization, which was later developed to produce Zermelo–Fraenkel set theory (ZF), which is now the most widely used Mathematical basic theory.