# Sentential Calculus for IT 101

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) <=> ¬Av¬B
• ¬(AvB) <=> ¬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