Over the course of these last three months I have had some pretty busy afternoons while studying for the certificate of Introduction to Mathematical Thinking, with the PhD in mathematics Keith Devlin, from Stanford University. In it I have discovered very interesting mathematical concepts, such as conditions of sufficiency and need, to which I already dedicated a post, or the existence quantifiers.
The only intention of creating these small summaries is, initially, to be able to better assimilate these concepts and leave some small and simple notes in case you need them in the future. However, since I was going to do them anyway, I decided to publish them on the blog, in case anyone needs a simple (introductory) explanation of these concepts.
Post content:
Existence Quantifiers in Mathematics
Mainly, when we talk about existence quantifiers in mathematics, we are going to find two specific symbols:
- The symbol "∃", which means "there is at least one case."
- And the symbol «∀», which means «for all cases».
There is also a third very useful symbol which is «∈», and means "belonging to a group."
With this simple explanation, let's dive a little deeper into the concepts of "existence quantifiers"; although if you don't yet know the conditions of sufficiency and need, you will need to understand them in order to understand some concrete examples of this explanation.
Utility of Existence Quantifiers
As far as I have been able to appreciate, its usefulness lies in be able to write abbreviated and very precise form concepts or phrases that would require a higher level of interpretation or reading if they were not written with mathematical symbols.
How to use them
To effectively write these existence quantifiers in mathematical language, you will always have to do so accompanied by a quantization domain, and to the left of a condition or characteristic. What does this mean?
What I mean is that you need to quantify something specific, and then make clear what characteristics or conditions it meets.
For example: ∀x ∈ N (x > 0), what does it mean "For all x that are a natural number, it is true that x is greater than zero«. (Taking into account that 0 is not a natural number).
The x can actually be anything: numbers, people, books... and different quantifiers and different quantization domains can also be combined.
For example: ∀x ∃y (x <= y) "For any number you choose, there is a number that is greater than or equal to that."
[In construction…]
Complicated Concepts of Existence Quantifiers in Mathematics
Negations of each Existence Quantifier
simple denial
Although it seems easy to interpret the denial of existence quantifiers within mathematics, it took me a while to understand it, and it is not as simple as denying the entire phrase.
For example: for the phrase: ∀x [ P(x) ] «Of all the x's, they all satisfy the property P«, you might think that its negation is: ∀x ¬[ P(x) ] «For all x, none of them have the property P«. However, his denial is the following: ∃x ¬[ P(x) ] «There exists an x that does not satisfy property P«. As you can see, this second denial also makes sense, since by finding a case that nullifies the first statement, you would be able to deny it; in fact, it makes the only logical sense that the denial of ∀x [ P(x) ], but it can be difficult to understand it that way.
In fact, ∀x ¬[ P(x) ] It is not the denial of ∀x [ P(x) ], but of ∃x [ P(x) ]. In case in the first sentence you say «No x satisfies property P«, in this last sentence you are saying «There is an x that does have the property P«. In this way you could affirm that the first sentence is false.
More complex denial
But the art of denying quantifiers of existence does not stop at what we have seen above. There are many more combinations that will blow our minds when trying to understand their negation. For example: ∃x [ P(x) ∧ ¬Q(x)], what does it mean: "There exists an x that has the property P and does not have the property Q«.
The way to negate this phrase is: ∀x [¬P(x) ∨ Q(x)], what does it mean: "All x's either do not satisfy P, or satisfy Q«. What logical sense does this make?
Well, everyone in the world, of course. If at the beginning you are saying that there is an x that satisfies one property and not the other, the logical way to deny it is to affirm that absolutely all In this way we would be achieving that the first statement could never be true, for any combination.
Furthermore, returning to the topic of the conditions of sufficiency and necessity, we also have that the denial of ∃x [P(x) ∧ ¬Q(x)] can be expressed as ∀x [P(x) ⇒ Q(x)], since taking into account that P implies Q, we find exactly the same condition as before: it cannot happen that P is fulfilled, and Q is not fulfilled, because again: P implies Q. In this way we observe how there are certain equivalences between logical conjunctions and conditions of implication and necessity.
Equivalences to take into account
As we have seen in the previous example, there are certain equalities and inequalities that it is convenient to know. These are:
Logical rules:
- By denying a ∃, will be transformed into a ∀, and vice versa.
- By denying a ∧, becomes a ∨, and vice versa,
- Two conditions with the form ¬P(x) ∨ Q(x) can also be expressed in the form P(x) ⇒ Q(x).
Careful with:
- Say ∀x [P(x) ∨ Q(x)] It is NOT the same as saying ∀x [P(x)] ∨ ∀x [Q(x)].
- Say ∃x [P(x) ∧ Q(x)] It is NOT the same as saying ∃x [P(x)] ∧ ∃x [Q(x)].
However:
- Say ∀x [P(x) ∧ Q(x)] YES it is equivalent to ∀x [P(x)] ∧ ∀x [Q(x)].
- Say ∃x [P(x) ∨ Q(x)] YES it is equivalent to ∃x [P(x)] ∨ ∃x [Q(x)].
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