[\<- 1.4](1.4.md)

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# 1.5 Nested Quantifiers p1

- Let the domain of x and y be the reals

|Logical Statement|Words|
|-----------------|-----|
|∀x(∀y(xy=xy))    |"For all x∈ℝ (is an element of real numbers) and all y∈ℝ, xy = yx"|
|∀y(∀x(xy=xy))    |"For all y∈ℝ (is an element of real numbers) and all x∈ℝ, xy = yx"|

- Both columns are true
- `∀x(∀y(xy=xy)) ≡ ∀y(∀x(xy=xy))`

Quantifiers **inside** the scope of other quantifiers are called **nested quantifiers**

- Note: `∀s ∀t P(s,t) ≡ ∀t ∀s P(s,t)

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- Ex. Let the domain of x and y be **positive reals**. Convert the logical statement to words and determine its truth value.
	1. `∀x Ǝy (x=y^2)`
		- "For all x>0, there exists y>0 s.t. (such that) x=y^2."
		- This statement is **True** since for any x, we would pick y=sqrt(x)
			- e.g. if x=4, then y=2
			- e.g. if x=5.72, then y=sqrt(5.72)
	2. `Ǝy ∀x (x=y^2)`
		- "There exists a number y>0 s.t. for all x>0, x=y^2."
		- This statement is **False** since once y is chosen, y^2 can only be one value. Therefore, only one value of x would satisfy x=y^2 (and not all x).
		- Note: `∀x Ǝy P(x,y)` is **not** equivalent to `Ǝy ∀x P(x,y)`

- Ex. Let the domain of x and y be integers. Convert the logical statement to words and determine its truth value.
	1. `Ǝx Ǝy (x^2 + y^2 = 3)`
		- In words: "There is an integer x and there is an integer y s.t. x^2 + y^2 = 3"
	2. `Ǝy Ǝx (x^2 + y^2 = 3)`
		- In words: "There is an integer y and there is an integer x s.t. x^2 + y^2 = 3"
	- Both of these examples have the same meaning
		- `Ǝx Ǝy P(x,y) ≡ Ǝy Ǝx P(x,y)`
	- These statements are **False** since no **integer** values of x and y can be plugged in to create a true statement

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**Careful**: `∀x(Ǝy P(y) -> ...)` is **not** equivalent to `∀x Ǝy(P(y) -> ...)`
- Be careful when moving quantifiers to make sure the meaning is the same

- Ex. Let P(Y) be "student y turns in their homework". Domain of y is students in our class. Let q be "Luvreet is happy"
	- `Ǝy P(y) -> q`
		- "If there is a student y who turns in their homework, then Luvreet is happy"
	- `Ǝy(P(y) -> q)`
		- "There is a student y s.t. if y turns in their homework, then Luvreet is happy"
	- These propositional functions are different the first one is basically saying, "Luvreet is happy when **any** student turns in their homework" but the second is basically saying, "Luvreet is happy when y turns in their homework"
	- A true statement would be: `Ǝy P(y) -> q ≡ ∀y(P(y) -> q)`

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