Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Mathematical Logic and Foundations → Propositional Logic ↓


Truth Tables


In the study of mathematical logic, truth tables are a powerful tool used to determine the validity of logical expressions. They provide a structured way of exploring how the truth values of propositions affect the truth value of a compound statement. By listing all possible scenarios and their consequences, truth tables offer a clear picture of logical relationships.

Introduction to propositional logic

Propositional logic, also known as propositional calculus, is a branch of mathematical logic that investigates how propositions relate to one another. Each proposition has a truth value; it can be true or false but not both. Propositions are the basic building blocks in this logic system, and they are often represented by variables such as P, Q, or R.

Basic logical operations

Before delving into truth tables, it is important to understand the basic logical operations that can be applied to propositions.

Coordinator

The conjunction of two propositions P and Q is noted as P ∧ Q The conjunction is true only if both P and Q are true. Otherwise, it is false.

    p | q | p ∧ q
    T | T | T
    T | F | F
    F | T | F
    F | F | F

Isolation

The disjunction of two propositions P and Q is noted as P ∨ Q The disjunction is true if at least one of P or Q is true. It is false only if both are false.

    p | q | p ∨ q
    T | T | T
    T | F | T
    F | T | T
    F | F | F

Denial

The negation operation reverses the truth value of a proposition P, denoted as ¬P or ~P. If P is true, then ¬P is false, and vice versa.

    p | ¬p
    T | F
    F | T

Conditional

The conditional operation (also called implication) is written as P → Q, which means “if P, then Q”. It is false only if P is true, and Q is false.

    p | q | p → q
    T | T | T
    T | F | F
    F | T | T
    F | F | T

Binary

The biconditional operation is represented by P ↔ Q and is true only if P and Q have the same truth value.

    p | q | p ↔ q
    T | T | T
    T | F | F
    F | T | F
    F | F | T

Construction of truth table

To construct a truth table for a given logical expression, follow these steps:

  • Identify the number of distinct variables in the expression.
  • Count the number of possible truth value combinations (2^n where n is the number of variables).
  • Create a table with a column for each variable and each operation.
  • List all possible combinations of truth values under the columns of variables.
  • Compute the truth value of the entire expression for each combination of truth values.

Example: Draw truth table for (P ∧ Q) → R

Consider an expression with three variables P, Q and R. We are interested in the operation (P ∧ Q) → R.

    p | q | r | p ∧ q | (p ∧ q) → r
    T | T | T | T | T
    T | T | F | T | F
    T | F | T | F | T
    T | F | F | F | T
    F | T | T | F | T
    F | T | F | F | T
    F | F | T | F | T
    F | F | F | F | T

Note how the truth values of P ∧ Q and (P ∧ Q) → R change according to the basic logical operations defined earlier. The expression is false only if P ∧ Q is true but R is false.

Complex expressions

Truth tables can be very large for complex expressions that involve a significant number of variables and operations. Nevertheless, they remain an invaluable tool for verifying the validity of logical statements.

Example: (P ∧ (Q ∨ R)) ↔ ¬(P → R)

Let's build a truth table for this more complex expression.

    p | q | r | q ∨ r | p ∧ (q ∨ r) | p → r | ¬(p → r) | (p ∧ (q ∨ r)) ↔ ¬(p → r)
    T | T | T | T | T | T | F | F
    T | T | F | T | T | F | T | T
    T | F | T | T | T | T | F | F
    T | F | F | F | F | F | T | F
    F | T | T | T | F | T | F | T
    F | T | F | T | F | T | F | T
    F | F | T | T | F | T | F | T
    F | F | F | F | F | T | F | T

This truth table shows the systematic approach needed to verify logical equivalences or demonstrate the tautology of complex expressions.

Importance of truth tables

Truth tables play an important role in various fields, such as computer science, artificial intelligence, and electrical engineering. They are used to design circuits, create algorithms, and verify the behavior of digital systems.

By understanding truth tables well, one can validate logical propositions, check equivalence between two expressions, and ensure that logical conclusions are true. This foundational knowledge is necessary to tackle more advanced topics in logic and mathematics.

Conclusion

Truth tables provide an impeccable way to discover fundamental operations and relationships in propositional logic. While they are simple tools, their application is profound across many topics. Through practice and continued application, the process of creating and interpreting truth tables becomes intuitive, allowing one to solve complex logic problems with ease.


Graduate → 8.1.1


U
username
0%
completed in Graduate

← Prev (8.1)
Propositional Logic
Next (8.1.2) →
Logical Equivalences

Comments 



Truth Tables

https://www.buddymath.com/g/8.1.1?bmal=en