# Introduction to set theory/Lecture 1

## Contents

## Introduction[edit]

We will start the course by introducing Propositional Logic. Even though, this is a set theory class, not a logic course, most of notations from the logic courses can be used in set theory. Furthermore, logic is important in various proofs we will encounter in this course.

## Notations[edit]

Here are the notations and what they mean:

Symbols | Meaning |
---|---|

and (conjunction) | |

or (nonexclusive disjunction) | |

not (negation) | |

if then/implies | |

if and only if |

## Truth Table[edit]

Truth tables are used to analyze formulae of propositional logic.

### Example[edit]

Truth table for

T | T | T | T |

T | F | T | T |

F | T | F | T |

F | F | T | T |

## Tautology[edit]

### Definition[edit]

A formula of propositional logic is a **tautology** if only T's occur in the column of the truth table.

### Examples[edit]

Truth table for

T | F | F | T |

F | T | T | T |

Truth table for

T | T | F | F | F | F | T |

T | F | T | F | F | F | T |

F | T | F | F | T | T | T |

F | F | T | F | T | T | T |

Truth table for

T | T | F | T | T | T |

T | F | F | F | F | T |

F | T | T | T | T | T |

F | F | T | T | T | T |

## Tautological Equivalence[edit]

### Definition[edit]

The proposition formulas and are **tautologically equivalent** if is a tautology.

### Examples[edit]

**Contraposition**: is tautologically equivalent to .

T | T | F | F | T | T | T |

T | F | T | F | F | F | T |

F | T | F | T | T | T | T |

F | F | T | T | T | T | T |

**de Morgan's Law I**: is tautologically equivalent to .

T | T | F | F | T | F | F | T |

T | F | F | T | T | F | F | T |

F | T | T | F | T | F | F | T |

F | F | T | T | F | T | T | T |

**de Morgan's Law II**: is tautologically equivalent to . *Truth table for Assignment #1*

## Related Resources[edit]

The materials in this course overlap with Introductory Discrete Mathematics for Computer Science, particularly Lesson 1.