<p>Yes, it is. Step by step:</p>
<p><span class="math-container">$\lnot \exists B \in A (x,y \in A \land x \in C(B))$</span> is equivalent to</p>
<p><span class="math-container">$\forall B \in A \lnot(x,y \in B \land x \in C(B))$</span> is equivalent to</p>
<p><span class="math-container">$\forall B \in A (x,y \notin B \vee x \notin C(B))$</span> is equivalent to</p>
<p><span class="math-container">$\forall B \in A (x,y \in B \implies x \notin C(B))$</span>.</p>
<p>Reference: <a href="https://www.amazon.com.br/How-Prove-Structured-Daniel-Velleman/dp/0521675995" rel="nofollow noreferrer">How to prove it, by Velleman</a></p>

CC BY-SA 4.0
2021-01-17T19:13:15.407Z
2021-01-19T20:05:39.570Z
2021-01-19T20:05:39.570Z
49328481
<p>I'm trying to negate that: <span class="math-container">$\exists B \in \mathcal{B}$</span> such that <span class="math-container">$x,y \in B$</span> and <span class="math-container">$x \in C(B)$</span>.</p>
<p>Looks that the negation is equivalent to: <span class="math-container">$\forall B \in \mathcal{B}(x,y \in B \implies x \not \in C(B))$</span>. Is it right?</p>

CC BY-SA 4.0
2021-01-19T22:21:50.593Z
This is more appropriate for math.se no? Whilst it fits into the context of economics I don't see how economics would help you prove this...
Brennan
CC BY-SA 4.0
2021-01-20T10:24:54.693Z
Once the appropriate meaning is attached to these letters it becomes an economics question. MWG is one of THE standard textbooks.
Bayesian
<microeconomics><mwg>
Is it enough to prove that $x \not \succsim^* y$?
49328481
# Is it enough to prove that $x \not \succsim^* y$?

(Asked by: username_0 on 2021-01-17)

I'm trying to negate that: $\exists B \in \mathcal{B}$ such that $x,y \in B$ and $x \in C(B)$.

Looks that the negation is equivalent to: $\forall B \in \mathcal{B}(x,y \in B \implies x \not \in C(B))$. Is it right?

**Tags:** microeconomics, mwg
> **Comments:**
> **username_1**: This is more appropriate for math.se no? Whilst it fits into the context of economics I don't see how economics would help you prove this...  
> **username_2**: Once the appropriate meaning is attached to these letters it becomes an economics question. MWG is one of THE standard textbooks.  

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## 1 answers

### Answer by username_0 on 2021-01-17. Score: 0 | Accepted answer

Yes, it is. Step by step:

$\lnot \exists B \in A (x,y \in A \land x \in C(B))$ is equivalent to

$\forall B \in A \lnot(x,y \in B \land x \in C(B))$ is equivalent to

$\forall B \in A (x,y \notin B \vee x \notin C(B))$ is equivalent to

$\forall B \in A (x,y \in B \implies x \notin C(B))$.

Reference: [How to prove it, by Velleman](https://www.amazon.com.br/How-Prove-Structured-Daniel-Velleman/dp/0521675995)


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