# Tagged Questions

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### Prove or Disprove the following statement. For any sets $A$, $B$, and $C$, we have $A \cup (B \& C) = (A\cup B) \cup (A\&C)$

Trying to figure this question out in my proofs class (tried venn-diagram the multiple set-notation signs are confusing me). Homework question in the fundamental sets unit.
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### Definition of $Z_m$ is $[n] = \{x | x \equiv n (\mod m)\}$?

Any help or sort of input on this question would help a great deal. Thanks Let $m\in N$. Recall for any integer $n \in Z$, the definition of $[n]$ in $Z_m$ is $[n] = \{x | x \equiv n (\mod m)\}$. ...
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### Let $H=\{2^m: m ∈ Z\}$ Where $m$ is any integer, and $a\sim b\Leftrightarrow a/b$ is an element of $H$.

Show that is an equivalence relation and describe the elements in the equivalence class $\operatorname{cl}(3)$. We're studying sets and equivalence in my mathematical proofs class. As this is a ...
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### Show that if S=a+b√2 : a,b are rational numbers and T=r+s√3 :r,s are rational numbers, then$S \cap T$ = rational

Someone please correct a formatting error in the problem [still a newbie] ; "S&T" (And = upside down U) Here's a bonus question that was on a test we received that I couldn't figure out. I'd ...
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### Find the number of integers between 1 and 100 that are divisible by exactly one of 3 and 4. [closed]

Homework question similar to my previous question but a little more specific. The last one I could follow with the hints but this one seems tougher, proofs class. Our current unit is sets, ...
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### Proof of de Morgans' law: For any sets $X$ and $Y$, $\overline{X\cup Y}= \overline{X}\cap\overline{Y}$ [duplicate]

For any sets $X$ and $Y$, $\overline{X\cup Y}= \overline{X}\cap\overline{Y}$ [Sorry, the hat should be a horizontal bar ] Same as the last two questions - this set theory stuff just isn't my walk in ...
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### Prove/Disprove: For any sets $X$ and $Y$, $\overline{X\cap Y} = \bar{X}\cup\bar{Y}$

Prove/Disprove: For any sets $X$ and $Y$, $\overline{X\cap Y} = \bar{X}\cup\bar{Y}$ Extra question in my proofs homework, similar to another question I posted, but not exactly the same I don't think. ...
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### True or false? $(X\setminus Y)\cup(Y\setminus Z)\cup(Z\setminus X) = X\cup Y\cup Z$, for any sets $X$, $Y$, $Z$.

Question in my proofs homework, I am not too familiar with sets. How would I start with this? Maybe contradiction?
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### Prove that $A\setminus (B\setminus C) = (A\setminus B) \cup (A\setminus C^c)$ for sets $A,\ B,\ C$ in some Universal Set $U$.

I'm working on this proof for some students I am tutoring and I've gotten a little stuck. I want to show them how to do a proof in complete, extravagant detail and get them familiar with ''element ...
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### Axiom of regularity and ordinal ranks

I am trying to prove that the following two statements are equivalent: Axiom of regularity $\forall x \exists \alpha (\alpha$ is an ordinal and $x \in V_\alpha)$ I believe I understand how to ...
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### Determining the properties for the relation over $P(\mathbb{N})$ where $ARB \iff A \cup B \in H$

I had two problems with this exercise: I don't know the universe for doing $\overline{A}$ (I'll show below). I couldn't show that it was transitive, although I'm fairly sure it is. Can you assist ...
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### How to approach proving $f^{-1}(B\setminus C)=A\setminus f^{-1}(C)$?

Let $A,B,C$ be sets such that $C\subseteq B$. Let $f: A \to B$ be a function. Prove that $f^{-1} (B\setminus C)=A\setminus f^{-1} (C).$ I really need help with this proof problem. I'm not sure ...
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### Proving a Property of a Set of Positive Integers

I have a question as such: A set $\{a_1, \ldots , a_n \}$ of positive integers is nice iff there are no non-trivial (i.e. those in which at least one component is different from $0$) solutions ...
The following is my attempt at one of my homework assignments. Let A, B, and C be sets. If the statement below is true, prove it. If false, give a counter example. A $\times$ (B $\cap$ C) = ...