## This article is a little introduction to sentential calculus for IT engineering students. I will try to explain it as correct and simple as possible.

Sentential calculus is the foundation for discrete and computational mathematics. It teaches you what is logical in a mathematical sense. Let’s begin with some terms.

### True or False

Every predicate must either equal true or false. Always. There is no null, like it should not exist for booleans.

Examples:

- 3+1 = 4 –> True
- I study IT –> True
- Unicorns exist –> False

### Sentential Connective

The sentential connectives connect two or more predicates as mentioned above. Let’s say we have the example

- A(x) := “x >2”

and

- B(x) := “x<10”

If we want to say x is between 2 and 10 you can do it with the AND connector. But more in this table.

Math | Informatics |

¬A | if(!A) |

A^B | if(A&&B) |

AvB | if(A||B) |

A=>B | if(!A||B), if(a) then b=true |

A<=>B | if((!A||B)(A||!B)) |

In this table I tried to match the connectives with a matchin programming pattern. Of course it’s only pseudo code.

#### Equivalence

A couple of rules regarding equal predicates:

- double negation leads to no negation: a != !false –> a = true
- A^B <=> B^A
- AvB <=> BvA
- (A^B)^C <=> A^(B^C)
- etc…

### De Morgan

The de morgan rules are actually pretty simple:

Every time you negate a sentential connective, you negate every predicate and change the connective sign.

Example:

- ¬(A
**^**B) <=> ¬A**v**¬B - ¬(A
**v**B) <=> ¬A**^**¬B

### Quantifiers

Quantifiers give you the ability to generate a predicate from other (multiple) predicates. Let’s say we have two predicates:

- A(x) := “x is a prime number”
- B := “There is a prime number that is a divisor of 24”

Or

- B:= “there is (at least) one x where A(x)” which is true

OR

- C:= ” every number x fulfills A(x)” which is wrong

#### List

Here’s a simpe list of quantifiers using natural numbers:

- ∀xA(x) means: For each x in N -> A(x) == true
- ∃xA(x) means: There is at least one x in N where A(x) == true

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