# Tagged Questions

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### Proving some properties of $\Bbb N$ without using recursion

I will try to prove that if $a, b, c \in \Bbb N$ and $a \in b \in c$, then $a \in c$ (the transitivity property). I will not use recursion or the replacement axiom. Actually we can proove in the same ...
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### Verification of Proof strategy

I am tasked with proving the following : $$A \cap B^c \subseteq (A \cap B)^c$$ I came up with the idea of using a combination of De Morgan's laws, rule simplification and rule of addition to prove ...
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### Proof makes sense?

$\exists! {A} \subset {Z}$ such that $A \cup B = A$, where $B$ is any subset of $Z$. Proof: Assume two such sets exist, $A_1$ and $A_2$ If $A_1 \cup B = A_1, \forall B \cup Z$, then $A_1 = Z$ If ...
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### Prove/disprove questions on isomorphism and embedding between order types

About the notations: Let $\lambda, q, z, \omega$ be the order types of the reals, rationals, integers and natural numbers respectively. The sign $=$ means there's isomorphism and $\le$ means ...
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### Showing that two intervals are equivalent.

Complete the proof that any two open intervals $(a, b)$ and $(c, d)$ are equivalent by showing that $f(x) = \frac{d-c}{b-a}(x-a) + c$ maps one to one and onto $(c,d)$. I showed one to one by saying ...
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### Velleman's How to prove it. Partial order proof.

Theorem: Suppose that $R$ is a partial order on $A$, $B_1 ⊆ A$, $B_2 ⊆ A$, $x_1$ is the least upper bound of $B_1$, and $x_2$ is the least upper bound of $B_2$. Prove that if $B_1 ⊆ B_2$ then ...
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### Prove if $a\in A$ is a maximum then $f(a)\in B$ is maximum and if $(A,\le_A)$ is totally ordered then $(B,\le_B)$ is totally ordered

Two ordered sets $(A,\le_A), (B,\le_B)$ and there's an isomorphic function $f:A\to B$ Prove if $a\in A$ is a maximum then $f(a)\in B$ is maximum. if $(A,\le_A)$ is totally ordered then ...
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### Would like to confirm answer (regarding sets)

As you might know from my precious questions, I am pretty weak with quantifiers. Below is my solution to the stated problem, if incorrect, could someone explain why? My attempted solution: ...
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### How to Prove it 4.1 ex.10

Prove that for any sets A, B, C, and D, if A × B and C × D are disjoint, then either A and C are disjoint or B and D are disjoint. Proof(someones). Suppose (A X B) and (C X D) are disjoint. Let (x,y) ...
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### Prove/disprove questions on equivalence relations and ordered sets

If $R$ is an equivalence relation and a partial order over $A \neq \emptyset$ then every equivalence class contain at least one element. If $(A,\le)$ an ordered set, and $a\in A$ is a single ...
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### Proof or find a counterexample:For all sets $A;B;C$ if $A\subseteq B,\ B\subseteq C,$ and $C\subseteq A,$ then $A=B=C.$

Proof or find a counterexample:For all sets $A;B;C$ if $A\subseteq B,\ B\subseteq C,$ and $C\subseteq A,$ then $A=B=C.$ My solution: True. Let $x\in A$, and since $A\subseteq B$ this implies that ...
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### Let $A =[a,b,c,f,g,i], B=[b,f,h]$ and $C = [a,k,l,m]$ Show that $\backslash$ is not associative

Question : Let $A =[a,b,c,f,g,i], B=[b,f,h]$ and $C = [a,k,l,m]$ Show that $\backslash$ is not associative by comparing $(A \backslash B) \backslash C$ with the set $A \backslash(B \backslash C)$. ...
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### Beginner proof of image of functions and functions of sets

This is the third time I got my proofs handed back from my teacher. She won't tell me what's wrong except I have to redo it. I am running out of luck and I need help towards the right direction! The ...
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### Please check these proofs for sets

I would appreciate the insight again for a couple of proofs since I'm learning. These are homework problems in so much as they are problems from the textbook. They are not required by my professor. ...
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### Problems with a proof that -in a linear order- a minimal element is the smallest element

I have a problem with a proof I found in Velleman's "How to prove it". This is sort of interesting, because it is the very first time I cannot see the structure of a proof presented in the book. The ...
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### for n∈N A(n)={x∈N | 0<= x <= n}, prove the following statements

please help me improve this proofes, or find a more formal mathematical version of them. N is the set of natural numbers N = {0,1,2,...,} for all $n∈N$, there is $A(n) = \{x∈N | 0\le x \le n\}$ ...
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### Prove the following sets equalities

I'm really struggling with proofes, please tell me if I'm correct and if there is a better way to prove (or disprove) the following: i) $(A \setminus B) \setminus B = A \setminus B$ My answer: ...
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### Proof: $a^2 - b^2 = (a-b)(a+b)$ holds $\forall a,b \in R$ iff R is commutative

We want to show that for some ring $R$, the equality $a^2 - b^2 = (a-b)(a+b)$ holds $\forall a,b \in R$ if and only if $R$ is commutative. Here's my proof --- I'm not sure if the first part stands ...
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### Prove or disprove $(A + B) \cap C = (A \cap C) +(B \cap C)$

Prove or disprove $(A + B) \cap C = (A \cap C) +(B \cap C)$ I want to disprove this statement. $(A+B)$ is the symmetric difference and has the form of $(A \cup B) \backslash (A \cap B)$ I am ...