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Sentential Calculus for IT 101

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) <=> ¬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

Published inMath for ITTutorials

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