You've uploaded a mathematics exercise sheet containing several problems across topics like set theory, relations, and basic algebra. Let's tackle each exercise:

Exercise 1: List the Elements of the Sets

1. $\{1, 2, 3, 4\}$: Elements are 1, 2, 3, 4.
2. $\{10, \{20\}, \{1, 2, 3\}, 3\}$: Elements are 10, $\{20\}$, $\{1, 2, 3\}$, and 3.
3. $\{5\}$: The only element is 5.
4. $\{\{5\}\}$: The element is $\{5\}$ which is a set itself.
5. $\emptyset$: This set is empty, no elements.
6. $\{x \in \mathbb{R} \mid x^2 - 2x = 0\}$: This describes the set of real numbers where $x^2 - 2x = 0$. Factoring gives $x(x - 2) = 0$, so the elements are $x = 0$ and $x = 2$.

Exercise 2: Prove Set Identity

We need to show $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$.

• LHS: $A \cap (B \cup C)$ includes all elements that are in both $A$ and either in $B$ or $C$.
• RHS: $(A \cap B) \cup (A \cap C)$ includes all elements that are either in both $A$ and $B$, or in both $A$ and $C$.

Using distributive laws of sets, it's evident that both expressions represent the same set of elements.

Exercise 3: Venn Diagrams for Subsets

• Venn Diagram: Represents subsets $A, B, C$ with $A \subseteq B$, $A \subseteq C$, and $B \cap C \subseteq A$. The overlapping regions need to reflect these subset relations accurately.

Exercise 4: Power Set and Set Differences

• $A = \{1, 2, 3\}$ and $B = \{1, 3\}$
• Power Set $P(A)$: $\{\emptyset, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}, \{1, 2, 3\}\}$
• $A \cap B = \{1, 3\}$
• $P(A - B) = P(\{2\}) = \{\emptyset, \{2\}\}$
• $P(A) - P(B)$: Contains all subsets of $A$ that are not subsets of $B$.

Exercise 5: Union of Sets and Power Sets

• Union: $s' = \{ \{1, 2, 3, 4\}, \{3, 4\}, \{2, 3, 5\} \}$
• Elements: Individual subsets listed, along with any elements derived from set operations.
• Existence of $S$: Looking for a set $S$ such that $s' = P(S)$ appears unlikely due to the non-systematic nature of the subsets in $s'$.

Exercise 6: Directed Graph Relations

• Directed Graph: Convert the graphical representation into a set of ordered pairs.
• $\{(x1, x2), (x1, x3), (x2, x3), (x2, x5), (x3, x5), (x5, x3)\}$

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Related Questions:

1. How do you prove set identities using element-wise proofs?
2. Can you explain more about the power set and its properties?
3. What is the significance of ordered pairs in the context of relations?
4. Can you detail the use of Venn diagrams in understanding complex set relations?
5. What are some practical applications of set theory in computer science?

Tip: Always check whether your sets contain other sets as elements when listing elements or performing set operations; this often causes confusion.