Let’s analyze and determine whether each of the given statements is true or false for subsets $A, B, C$ of a universal set $U$:

(a) $(A \subseteq B \land A \subseteq C) \implies A \subseteq B \cap C$

• Analysis:

  • $A \subseteq B$ means every element of $A$ is in $B$.
  • $A \subseteq C$ means every element of $A$ is in $C$.
  • The intersection $B \cap C$ consists of elements common to both $B$ and $C$.
  • Therefore, if $A \subseteq B$ and $A \subseteq C$, then every element of $A$ is also in $B \cap C$.

• Conclusion: True.

(b) $A \subseteq B \iff A \cap B = A$

• Analysis:

  • $A \subseteq B$: Every element of $A$ is in $B$.
  • $A \cap B = A$: The intersection of $A$ and $B$ contains all elements of $A$.
  • If $A \subseteq B$, then all elements of $A$ are in $B$, so $A \cap B = A$.
  • Conversely, if $A \cap B = A$, then all elements of $A$ must be in $B$, so $A \subseteq B$.

• Conclusion: True.

(c) $A^c \setminus B^c = B \setminus A$

• Analysis:

  • $A^c$: The complement of $A$, which consists of all elements not in $A$.
  • $B^c$: The complement of $B$, which consists of all elements not in $B$.
  • $A^c \setminus B^c$: Elements that are in $A^c$ but not in $B^c$.
    • This is equivalent to elements not in $A$ but in $B$.
  • $B \setminus A$: Elements that are in $B$ but not in $A$.

  Since both expressions represent the set of elements in $B$ but not in $A$, the equality holds.

• Conclusion: True.

(d) $A \subseteq B \implies A \cup (B \setminus A) = B$

• Analysis:

  • $A \subseteq B$: Every element of $A$ is in $B$.
  • $B \setminus A$: Elements in $B$ but not in $A$.
  • $A \cup (B \setminus A)$: The union of $A$ and the elements of $B \setminus A$.
    • This covers all elements of $A$ (which are already in $B$) and all elements in $B$ but not in $A$.
  • Thus, $A \cup (B \setminus A)$ equals $B$.

• Conclusion: True.

Final Results:

(a) True
(b) True
(c) True
(d) True

Do you need further clarifications on any of these proofs? Let me know if you’d like detailed steps for any specific statement.

Related Questions:

1. What is the definition of set complement, intersection, and union?
2. Can you explain the concept of logical equivalence using set theory?
3. What happens if we replace $\subseteq$ with $\subset$ in statement (b)?
4. How would the proofs change if the universal set $U$ is finite?
5. Can you derive other similar identities involving set difference and complements?

Tip:

Set operations such as union, intersection, and difference follow logical rules that closely align with Boolean algebra. Practice simplifying these operations using Venn diagrams for clarity.