3
votes
1answer
76 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
41 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
72 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
28 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
78 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
486 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
283 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
474 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 ...