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This is the problem that I've tried to solve. Let me know if I was right. It was written originally in italian, but I don't know how translate some mathematical words in english, so sorry in advice.

Given $S = \{1, 2, 3\}$, and the following function $f:X\in\mathbb{P}(S)\longmapsto X\backslash\{2\} \in \mathbb{P}(S)$

  1. Talk about injectivity and surjectivity
  2. Given the equivalence relation $\mathcal{R}_f$ induced by $f$ defined as following: $X \mathcal{R}_f Y \Longleftrightarrow f(X)=f(Y)$ describe every equivalence class in $\mathcal{R}_f$.
  3. Define in $\mathbb{P}(S)\backslash\{\varnothing\}$ the following order relation $ X \quad \Sigma \quad Y \Leftrightarrow X =Y \quad \text{or}\quad \max(X)<\max(Y).$ Determine in $\{\mathbb{P}(S)\backslash\{\varnothing\}, \Sigma\}$ min, max, maximal and minimal element.
  4. Is $\{\mathbb{P}(S)\backslash\{\varnothing\}, \Sigma\}$ a totally ordered set?

This is my solution:

  1. Not injective. $ \{1,3\}\neq\{1,2,3\}\quad \text{but} \quad f(\{1,3\})=f(\{1,2,3\}) $ Not surjective. (e.g. $ \{1,2\}\in B$ have not corresponding element in $A$ )
  2. Four equivalence classes: $[\{1\}]_{\mathcal{R}_f}=\{\{1\},\{1,2\}\}$ $[\{3\}]_{\mathcal{R}_f}=\{\{3\},\{2,3\}\}$ $[\{\varnothing\}]_{\mathcal{R}_f}=\{\{2\},\{\varnothing\}\}$ $[\{1,3\}]_{\mathcal{R}_f}=\{\{1,3\},\{1,2,3\}\}$

Am I right? How can I start to resolve 3rd and 4th point?

Best regard

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    @Neal, I have $\mathbb{P}(S)$ that looks like $\{\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\},\{\varnothing\},\}$. Final order should be something like {1}--{2}--{3}--{1,2}--{1,2,3}. Then {1} min, {1,2,3} max and no minimal or maximal element, is that true?2012-01-10

1 Answers 1

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This CW answer intends to remove this question from the Unanswered queue.


Your answers for 1. and 2. are correct.

Regarding 3., Neal's advice is good:

For 3, just write down all the elements of $\Bbb P(S)$ and draw a digraph with an arrow from $X$ to $Y$ exactly when $X\Sigma Y$. This should make the max, min, maximal, minimal elements obvious.

Regarding 4., André Nicolas's remark should do the trick:

On 4: If I understand the problem correctly, let $X = \{1,3\}$ and $Y=\{2,3\}$. then neither $X\;\Sigma\;Y$ nor $Y\;\Sigma\;X$ is true. So the order cannot be total.