To prove these set-theoretic properties, let's carefully analyze each statement:

(i) A - A = Φ

• Proof:
  • The difference $A - A$ means all elements in $A$ that are not in $A$. Since every element of $A$ is in $A$, there are no elements left, hence $A - A = \emptyset$.

(ii) A - B

• Clarification Required:
  • This seems incomplete. Did you mean to prove something specific about $A - B$? For example, $A - B = A \setminus B$?

(iii) A = U - A'

• Proof:
  • Here, $U$ is the universal set, and $A'$ denotes the complement of $A$ in $U$. The difference $U - A'$ consists of all elements in $U$ that are not in $A'$. By the definition of a complement, $A'$ consists of all elements not in $A$. Thus, $U - A'$ is precisely $A$, so $A = U - A'$.

(iv) A - Φ = A

• Proof:
  • The difference $A - \emptyset$ means all elements in $A$ that are not in $\emptyset$. Since $\emptyset$ contains no elements, $A - \emptyset = A$.

(v) A - B = B - A \text{ if and only if } A = B

• Proof:
  • (⇒ direction): Suppose $A - B = B - A$. Consider any element $x \in A$. Since $A - B = B - A$, $x$ must also belong to $B$, otherwise it would be in $A - B$ but not in $B - A$. Similarly, any element $x \in B$ must belong to $A$. Thus, $A = B$.
  • (⇐ direction): If $A = B$, then clearly $A - B = B - A = \emptyset$.

(vi) A - B = Φ \text{ if and only if } A \subseteq B

• Proof:
  • (⇒ direction): Suppose $A - B = \emptyset$. This means there are no elements in $A$ that are not in $B$, implying $A \subseteq B$.
  • (⇐ direction): If $A \subseteq B$, then all elements of $A$ are in $B$, so there are no elements left in $A - B$, hence $A - B = \emptyset$.

(vii) A - B = A \text{ if and only if } A \cap B = Φ

• Proof:
  • (⇒ direction): Suppose $A - B = A$. This implies no elements from $A$ are removed when subtracting $B$, which means $A$ and $B$ have no elements in common, i.e., $A \cap B = \emptyset$.
  • (⇐ direction): If $A \cap B = \emptyset$, then no elements from $A$ are in $B$, so subtracting $B$ from $A$ has no effect, hence $A - B = A$.

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Do you want more details or have any questions?

5 Related Questions:

1. Can you prove the distributive property of sets, $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$?
2. What is the relationship between $A \cap A'$ and $\emptyset$?
3. How does De Morgan's Law relate to the complements of union and intersection of sets?
4. Can we generalize the property $A - B = \emptyset$ to any arbitrary number of subsets?
5. How does the concept of symmetric difference relate to the proof of $A - B = B - A$?

Tip:

When dealing with set operations, Venn diagrams are an excellent tool for visualizing and understanding relationships between sets.