Table of Contents

# Fuzzy Logic – Conventional Fuzzy Refresher

Logic, which was initially simply the examine of what distinguishes sound argument from unsound argument, has now developed into a strong and rigorous system whereby true statements might be found, given different statements which might be already recognized to be true.

### Predicate Logic

This logic offers with predicates, that are propositions containing variables.

A predicate is an expression of a number of variables outlined on some particular area. A predicate with variables might be made a proposition by both assigning a price to the variable or by quantifying the variable.

Following are a couple of examples of predicates −

- Let E(x, y) denote “x = y”
- Let X(a, b, c) denote “a + b + c = 0”
- Let M(x, y) denote “x is married to y”

### Propositional Logic

A proposition is a group of declarative statements which have both a reality worth “true” or a reality worth “false”. A propositional consists of propositional variables and connectives. The propositional variables are dented by capital letters (A, B, and many others). The connectives join the propositional variables.

A couple of examples of Propositions are given beneath −

- “Man is Mortal”, it returns reality worth “TRUE”
- “12 + 9 = 3 – 2”, it returns reality worth “FALSE”

The next just isn’t a Proposition −

**“A is lower than 2”**− It’s as a result of except we give a particular worth of A, we can’t say whether or not the assertion is true or false.

### Connectives

In propositional logic, we use the next 5 connectives −

- OR (∨∨)
- AND (∧∧)
- Negation/ NOT (¬¬)
- Implication / if-then (→→)
- If and provided that (⇔⇔)

### OR (∨∨)

The OR operation of two propositions A and B (written as A∨BA∨B) is true if at the very least any of the propositional variable A or B is true.

The reality desk is as follows −

A | B | A ∨ B |
---|---|---|

True | True | True |

True | False | True |

False | True | True |

False | False | False |

### AND (∧∧)

The AND operation of two propositions A and B (written as A∧BA∧B) is true if each the propositional variable A and B is true.

The reality desk is as follows −

A | B | A ∧ B |
---|---|---|

True | True | True |

True | False | False |

False | True | False |

False | False | False |

### Negation (¬¬)

The negation of a proposition A (written as ¬A¬A) is fake when A is true and is true when A is fake.

The reality desk is as follows −

A | ¬A |
---|---|

True | False |

False | True |

### Implication / if-then (→→)

An implication A→BA→B is the proposition “if A, then B”. It’s false if A is true and B is fake. The remainder circumstances are true.

The reality desk is as follows −

A | B | A→B |
---|---|---|

True | True | True |

True | False | False |

False | True | True |

False | False | True |

### If and provided that (⇔⇔)

A⇔BA⇔B is a bi-conditional logical connective which is true when p and q are identical, i.e., each are false or each are true.

The reality desk is as follows −

A | B | A⇔B |
---|---|---|

True | True | True |

True | False | False |

False | True | False |

False | False | True |

Properly Shaped Formulation

Properly Shaped Formulation (wff) is a predicate holding one of many following −

- All propositional constants and propositional variables are wffs.
- If x is a variable and Y is a wff, ∀xY and ∃xY are additionally wff.
- Reality worth and false values are wffs.
- Every atomic system is a wff.
- All connectives connecting wffs are wffs.

## Quantifiers

The variable of predicates is quantified by quantifiers. There are two kinds of quantifier in predicate logic −

- Common Quantifier
- Existential Quantifier

### Common Quantifier

Common quantifier states that the statements inside its scope are true for each worth of the particular variable. It’s denoted by the image ∀.

**∀xP(x)** is learn as for each worth of x, P(x) is true.

**Instance** − “Man is mortal” might be reworked into the propositional kind ∀xP(x). Right here, P(x) is the predicate which denotes that x is mortal and the universe of discourse is all males.

### Existential Quantifier

Existential quantifier states that the statements inside its scope are true for some values of the particular variable. It’s denoted by the image ∃.

**∃xP(x)** for some values of x is learn as, P(x) is true.

**Instance** − “Some individuals are dishonest” might be reworked into the propositional kind ∃x P(x) the place P(x) is the predicate which denotes x is dishonest and the universe of discourse is a few folks.

## Nested Quantifiers

If we use a quantifier that seems inside the scope of one other quantifier, it’s referred to as a nested quantifier.

**Instance**

- ∀ a∃bP(x,y) the place P(a,b) denotes a+b = 0
- ∀ a∀b∀cP(a,b,c) the place P(a,b) denotes a+(b+c) = (a+b)+c

**Be aware** − ∀a∃bP(x,y) ≠ ∃a∀bP(x,y)