[Maths Class Notes] on Mathematical Logic Pdf for Exam

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:

1. Set Theory

2. Recursion Theory

3. Model Theory

4. 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:

1. Conjunction or (AND)

2. Disjunction or (OR)

3. 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.