In discrete mathematics, a tautology is a compound statement that is always true, regardless of the truth values of its individual components. It is a fundamental concept in propositional logic, used to verify logical expressions and implications. Tautologies play a crucial role in constructing proofs and understanding logical consistency.
What is Proposition?
The meaning of proposition in literature is an idea, a plan an offer, or a suggestion that can be proved True or False. The same goes for mathematical propositions. They are declarative sentences that can be True or False. Propositions are the fundamental building blocks of logic.
Examples:
1. The magnetic lines emerge from the North and merge into South pole.
2. 2 + 1 = 3
3. 'p' is a vowel.
All of the above three sentences are proper propositions, where the first two are True and the third one is False.
Tautology
Propositional logic is said to be a tautology if it is always true regardless of the true/false of the atomic formulas. A tautology is always “True". To check whether a given logic is a tautology or not, we often use the truth table method. However, the truth table method is not effective when the logic contains several atomic formulas.
Example:
Odd number = A
Even number = B
1. If we add one odd number and one even number then we get odd number.
Converting statement-1 into mathematical logic:
A ∧ B ⇒ A
Let's prove that the above logic is a tautology. In order to build the truth table, we need to convert the logic statements into a clausal form.
The truth table of A ∧ B ⇒ A, clausal form is: ￢(A ∧ B)∨A
| A | B | (A ∧ B) | ￢(A ∧ B) | ￢(A ∧ B)∨A |
|---|
| T | T | T | F | T |
| T | F | F | F | T |
| F | T | F | T | T |
| F | F | F | T | T |
All entries are True, irrespective of the True/False values of atomic literals. So, this is a tautology.
Examples of Tautology with logic symbols:
- ￢A∨A
- (P∨Q)⇒(P∨Q)
A mathematical sentence consisted of logic. A proposition is either True or False. The proposition is made up of mathematical logic. Various propositional logics are given below in their priority order:
- Negation (No)
- Conjunction (and)
- Disjunction (or)
- Implication (⇒)
- Equivalence (⇔)
Tautology- A proposition that is always true. The truth table is evaluated for the given proposition and if in every case the result is True, then that proposition is called Tautology.
Truth Table
It is a table that gives the output of the propositional logic against each input component. The result is binary, either True or False for each row of inputs.
Problems on Tautology
Find if the given propositional logic is a tautology or not.
1) P
Truth table:
The truth table of P contains a False value. Thus, it can not be a tautology.
2) P⇒P
We shall draw the truth table for this proposition.
Implication:
P⇒Q =￢P∨Q
The simplified expression of the given proposition is: ￢P∨P
Truth Table:
The truth table of ￢P∨P consists only of True values. Therefore, P⇒P is a tautology.
3) (P ⇒ P) ⇒ P
We shall draw the truth table for this proposition.
Implication:
P⇒Q=￢P∨Q
The simplified expression of the given proposition is:
(￢P∨P) ⇒ P
￢(￢P∨P)∨P
(￢(￢P) ∧ ￢P)∨P {By Demorgan's Law}
(P ∧ ￢P) ∨P
(P ∧ ￢P)= False {Complement laws: – P∧￢P=F }
False ∨ P = P {Absorption law}
Thus, (P ⇒ P) ⇒ P is equivalent to P. We have already solved this in problem-1.
Therefore, this is not a Tautology.
4) (p → q) → [(p → q) → q]
Solving: (p → q) = ￢p∨q {Implication}
Solving: [(p → q) → q]
= [(￢p∨q) → q]
= [￢(￢p∨q)∨q]
= [(￢(￢p)∧￢q)∨q] {Demorgan's Law}
= [(p∧￢q)∨q] {Involution law}
= [(p∨q)∧(￢q∨q)] {Distributive law}
= [(p∨q)∧T] {Complement law}
= (p∨q) {Absorption law}
Solving (p → q) → [(p → q) → q]
￢(￢p∨q)∨(p∨q)
[￢(￢p)∧(￢q)]∨(p∨q){Demorgan's Law}
(p∧￢q)∨(p∨q) {Involution law}
Thus, final expression is: (p∧￢q)∨(p∨q)
Truth Table:
| p | q | ￢q | (p∧￢q) | (p∨q) | (p∧￢q)∨(p∨q) |
|---|
| T | T | F | F | T | T |
| T | F | T | T | T | T |
| F | T | F | F | T | T |
| F | F | T | F | F | F |
Since there is a False entry in the truth table, it implies it is not a Tautology.
5) ((P⇒Q)∧P)⇒Q
Solving (P⇒Q): ￢P∨Q
Solving ((P⇒Q)∧P): ((￢P∨Q)∧P)
= (￢P∧P)∨(Q∧P) {Distributive Law}
= (F)∨(Q∧P) {Complement Law}
= (Q∧P) {Absorption Law}
Solving ((P⇒Q)∧P)⇒Q: (Q∧P)⇒Q
= ￢(Q∧P)∨Q
= (￢Q∨￢P)∨Q {Demorgan Law}
= (￢Q∨Q)∨￢P) {Associative Law}
= T∨(￢P) {Complement Law}
= T {Absorption Law}
Final CNF is: True
Here, no need of finding the Truth Table. Given logic is a tautology.
Related Articles:
Conclusion
Understanding and identifying tautologies are essential skills in discrete mathematics and logic. They help ensure the correctness and consistency of logical statements and are widely used in mathematical proofs and algorithm design. By practicing problems related to tautologies, students can enhance their logical reasoning and problem-solving abilities.
Practice Problems on Tautology
1. Verify if p ∨ ¬p is a tautology.
2. Check if p ∧ ¬p is a tautology.
3. Determine if q → (p ∨ q) is a tautology.
4. Test if (p ∨ q) ∧ ¬p ∧ ¬q is a tautology.
5. Validate if (p ∧ q) → p is a tautology.
6. Determine if (p → q) ∨ (q → p) is a tautology.
7. Check if [(p → q) ∧ (q → r)] → (p → r) is a tautology.
8. Verify if p → (q → p) is a tautology.
9. Test if (p ∧ q) ∨ (¬p ∨ ¬q) is a tautology.
10. Determine if [(p ∨ q) ∧ ¬p] → q is a tautology.
11. Check if (p → q) → [(¬q) → (¬p)] is a tautology.
12. Verify if [p ∧ (p → q)] → q is a tautology.
13. Test if (p ∨ q) → (q ∨ p) is a tautology.
14. Determine if [(p → q) ∧ (r → s) ∧ (p ∨ r)] → (q ∨ s) is a tautology.
15. Check if [p ∨ (¬p ∧ q)] ↔ (p ∨ q) is a tautology.
Similar Reads
Engineering Mathematics Tutorials Engineering mathematics is a vital component of the engineering discipline, offering the analytical tools and techniques necessary for solving complex problems across various fields. Whether you're designing a bridge, optimizing a manufacturing process, or developing algorithms for computer systems,
3 min read
Linear Algebra
MatricesMatrices are key concepts in mathematics, widely used in solving equations and problems in fields like physics and computer science. A matrix is simply a grid of numbers, and a determinant is a value calculated from a square matrix.Example: \begin{bmatrix} 6 & 9 \\ 5 & -4 \\ \end{bmatrix}_{2
3 min read
Row Echelon FormRow Echelon Form (REF) of a matrix simplifies solving systems of linear equations, understanding linear transformations, and working with matrix equations. A matrix is in Row Echelon form if it has the following properties:Zero Rows at the Bottom: If there are any rows that are completely filled wit
4 min read
Eigenvalues and EigenvectorsEigenvalues and eigenvectors are fundamental concepts in linear algebra, used in various applications such as matrix diagonalization, stability analysis and data analysis (e.g., PCA). They are associated with a square matrix and provide insights into its properties.Eigen value and Eigen vectorTable
10 min read
System of Linear EquationsA system of linear equations is a set of two or more linear equations involving the same variables. Each equation represents a straight line or a plane and the solution to the system is the set of values for the variables that satisfy all equations simultaneously.Here is simple example of system of
5 min read
Matrix DiagonalizationMatrix diagonalization is the process of reducing a square matrix into its diagonal form using a similarity transformation. This process is useful because diagonal matrices are easier to work with, especially when raising them to integer powers.Not all matrices are diagonalizable. A matrix is diagon
8 min read
LU DecompositionLU decomposition or factorization of a matrix is the factorization of a given square matrix into two triangular matrices, one upper triangular matrix and one lower triangular matrix, such that the product of these two matrices gives the original matrix. It is a fundamental technique in linear algebr
6 min read
Finding Inverse of a Square Matrix using Cayley Hamilton Theorem in MATLABMatrix is the set of numbers arranged in rows & columns in order to form a Rectangular array. Here, those numbers are called the entries or elements of that matrix. A Rectangular array of (m*n) numbers in the form of 'm' horizontal lines (rows) & 'n' vertical lines (called columns), is calle
4 min read
Sequence & Series
Calculus
Limits, Continuity and DifferentiabilityLimits, Continuity, and Differentiation are fundamental concepts in calculus. They are essential for analyzing and understanding function behavior and are crucial for solving real-world problems in physics, engineering, and economics.Table of ContentLimitsKey Characteristics of LimitsExample of Limi
10 min read
Cauchy's Mean Value TheoremCauchy's Mean Value theorem provides a relation between the change of two functions over a fixed interval with their derivative. It is a special case of Lagrange Mean Value Theorem. Cauchy's Mean Value theorem is also called the Extended Mean Value Theorem or the Second Mean Value Theorem.According
7 min read
Taylor SeriesA Taylor series represents a function as an infinite sum of terms, calculated from the values of its derivatives at a single point.Taylor series is a powerful mathematical tool used to approximate complex functions with an infinite sum of terms derived from the function's derivatives at a single poi
8 min read
Inverse functions and composition of functionsInverse Functions - In mathematics a function, a, is said to be an inverse of another, b, if given the output of b a returns the input value given to b. Additionally, this must hold true for every element in the domain co-domain(range) of b. In other words, assuming x and y are constants, if b(x) =
3 min read
Definite Integral | Definition, Formula & How to CalculateA definite integral is an integral that calculates a fixed value for the area under a curve between two specified limits. The resulting value represents the sum of all infinitesimal quantities within these boundaries. i.e. if we integrate any function within a fixed interval it is called a Definite
8 min read
Application of Derivative - Maxima and MinimaDerivatives have many applications, like finding rate of change, approximation, maxima/minima and tangent. In this section, we focus on their use in finding maxima and minima.Note: If f(x) is a continuous function, then for every continuous function on a closed interval has a maximum and a minimum v
6 min read
Probability & Statistics
Mean, Variance and Standard DeviationMean, Variance and Standard Deviation are fundamental concepts in statistics and engineering mathematics, essential for analyzing and interpreting data. These measures provide insights into data's central tendency, dispersion, and spread, which are crucial for making informed decisions in various en
10 min read
Conditional ProbabilityConditional probability defines the probability of an event occurring based on a given condition or prior knowledge of another event. It is the likelihood of an event occurring, given that another event has already occurred. In probability, this is denoted as A given B, expressed as P(A | B), indica
12 min read
Bayes' TheoremBayes' Theorem is a mathematical formula used to determine the conditional probability of an event based on prior knowledge and new evidence. It adjusts probabilities when new information comes in and helps make better decisions in uncertain situations.Bayes' Theorem helps us update probabilities ba
13 min read
Probability Distribution - Function, Formula, TableA probability distribution is a mathematical function or rule that describes how the probabilities of different outcomes are assigned to the possible values of a random variable. It provides a way of modeling the likelihood of each outcome in a random experiment.While a Frequency Distribution shows
13 min read
Covariance and CorrelationCovariance and correlation are the two key concepts in Statistics that help us analyze the relationship between two variables. Covariance measures how two variables change together, indicating whether they move in the same or opposite directions. Relationship between Independent and dependent variab
6 min read
Practice Questions