In this tutorial we going to learn propositional logic final chapter, this write consist of quantifiers, open sentence, and truth type. This article is final for propositional logic after that we going to look at clusters, and if you don’t read the first and second article, you should read to prepare.
What Is Open Sentence?
These are propositions that contain at least one variable and that this variable affects the result. In short, the variable should affect the whole result. They can be 1, 2, or 3.
p(x): "x + 12 < 9 , x C Z"
The proposition here is wrong. The reason for this is that the given values may be greater than 9 because the variable x is an integer. Let’s look at the sign of a few sets of numbers.
Z -> Integer N -> Natural Number R -> Reel Number Q -> Rational numbers
Let’s practice understanding open sentences after that we going to look at quantifiers.
More Example For Open Sentence
p(x): "4x^2 + 8 > 7 , x C N"
This proposition is a correct proposition because no matter what number you bring in, the result will always be greater than 7. Remember, there are no negative numbers in the set of natural numbers.
p(y): "(y * 12) ^ 2 = 1 , x C Z"
This proposition is also true, multiplying y by 12, but if the power of all integers is 0, the result is 1.
Quantifiers In Propositional Logic
It is used to indicate the abundance of the variable brought before it, There are two types, they are as “some” and “for each”. the symbol is used ∀ for “for each”, this is ∃ used for some.
p(x): "∃x > 2 , x C Z" -> True Statement
p(y): "∀x > 2 , x C Z" -> False Statement
The reason the upper one is correct and the lower one is false, some numbers are greater than 2, but in the 2nd proposition every proposition is tried, and not every proposition is greater than 2. Let’s pass on the types of the proposition.
Types of Proposition
In this chapter we will learn about the types of correct propositions, yes you heard right, there are also types of correct propositions. We divide these types into tautology, contradiction, and contingency.
Contradiction = Propositions that are always wrong are called contradictions. Contingency = Propositions that are neither tautological nor contradictory are called contingency. Tautology = The proposition is always correct and is called a tautology.
Examples Types Of Propositions
p(x): "12 > 1" -> Tautology p(y): "2 * 2 > 5" -> Contradiction p(z): "p v q" -> Contingency
Yes, we solved our last examples and finished the subject of propositional logic. We will look at clusters in the other math tutorial.
Congratulations, You Finished To Propositional Logic In Math!