0
votes
1answer
35 views

Axiom of Unions and its use of the existential quantifier

I'm reading Halmos's Naive Set Theory, and right now I'm on the section about the axiom of unions. As stated in the book, the axiom reads: For every collection of sets there exists a set that ...
3
votes
1answer
79 views

Semantics and Logical structure in Definitons

Continuation of Free and bound variables in "if" statements definitions: A number is even if it is divisible by $2$. The number is even if it is divisible by $2$. Is the usage of the ...
1
vote
1answer
44 views

Logical structure of definitions

Here are some "concepts" that are confusing me: The assertion $x>2$ is true when we replace $x$ by a number greater than $2$($3$ for example) and false if we replace it by $1$. But the assertion ...
3
votes
3answers
62 views

Truth set of a Universal Quantifier and Family of sets

The definition of an intersection of family($F$) is: $$\cap F=\bigl\{x\mid\forall A(A\in F \implies x\in A)\bigr\} $$ If my understanding serves me correctly this notation means that all the $x$ ...
3
votes
1answer
73 views

Definition of intersection of a family of sets

The definition of an intersection of family($F$) is: $$\cap F=\bigl\{x\mid\forall A: (A\in F \implies x\in A)\bigr\} $$ If my understanding serves me correctly this notation means that all the $x$ ...
1
vote
1answer
30 views

Set Theory Elementhood Notation

From How to Prove it: Given $A=\{n^2|n \in N\}$ where $N$ is the set of all natural numbers. I want to express A in terms of elementhood test notation. Velleman says $A=\{x| \exists n \in N ...
2
votes
3answers
79 views

First order logic. Describe that a set has more than 2 elements.

I would like to describe that a set has at least 3 elements using first order logic, would this be a valid way to do that? $\forall x\exists y\exists z(\neg(x=y)\wedge\neg(x=z)\wedge\neg(y=z))$ I ...
1
vote
2answers
40 views

Negating statements / Finding $(A \cap B)',A \oplus B$ if $A=\{x \in\Bbb R \mid -3\le x\le0\}$ and $B=\{x \in \Bbb R\mid -1 < x < 2\}$

I am a bit new on this field and I am trying to solve some questions. I don't really think they are hard but there are some key points that I don't get it or I am stuck. Lets see. 1) Write the ...
1
vote
1answer
69 views

What is the logical interpretation of this set?

$\{f \in C : f(x)>d$ for each $x$ for some $d\}$ Do you read the above set as "the set of functions in $C$ such that there exists $d$ such that for each $x, f(x)>d$" or do you read the $d$ as ...
2
votes
2answers
75 views

Getting weird results

Consider two nonempty sets $ S $ and $ T $, $S\subseteq T$. We can write: $$\forall x (x\in S \Rightarrow x\in T) $$ Knowing, that $ p\Rightarrow q $ is equivalent to $\neg (p \wedge \neg q) $, we ...
1
vote
2answers
525 views

Learning math symbols

I am taking linear algebra and none of this stuff is expained. I found this helpful link http://www.math.ucla.edu/~pskoufra/M115A-Notation.pdf but it is missing a lot of what I need to know. Just ...
9
votes
6answers
285 views

Why is “for all $x\in\varnothing$, $P(x)$” true, but “there exists $x\in\varnothing$ such that $P(x)$” false? [duplicate]

There exists an $X\in A$ such that $P(X)$. When $A$ is the empty set then this statement is false because there is nothing in $A$ that when plugged in for $X$, makes $P(X)$ come out True. However, ...
0
votes
1answer
43 views

Is it true that $\forall A\in \mathcal{F} P(A) \wedge\forall B\in\mathcal{G} P(B) \leftrightarrow\forall C\in(\mathcal{F}\cup\mathcal{G})P(C)$

While working on proof for $(\cap\mathcal{F})\cap(\cap\mathcal{G})=\cap(\mathcal{F}\cup\mathcal{G})$ I came up with this equivalence: $$\forall A\in \mathcal{F} P(A) \wedge\forall B\in\mathcal{G} ...
1
vote
2answers
161 views

Suppose that $ A \subseteq \mathscr{P}(A)$. Prove $\mathscr{P}(A) \subseteq\mathscr{P}(\mathscr{P}(A)) $

Well i've been having problems trying to prove that $\mathscr{P}(A) \subseteq\mathscr{P}(\mathscr{P}(A)) $ if $ A \subseteq \mathscr{P}(A)$ What i need is to get a proof by using quantifiers
2
votes
1answer
479 views

Free variables, bound variables and quantification - explanation

I'm trying to learn something about mathematical logic. I think, I partly understand definitions of free variables, bound variables and quantification, but it's not easy topic for me and it's not ...