Evaluate the truthfulness of the following set theory statements: 1. A - B ⊆ A 2. B - A ⊆ B 3. If A = B, then A + B = ∅ 4. A - B ≠ B - A

$A - B \subseteq A$

• Explanation: The set $A - B$ (also written as $A \setminus B$) contains all elements that are in set $A$ but not in set $B$. By definition, every element in $A - B$ must be an element of $A$. Therefore, $A - B$ is a subset of $A$.
• Truth Value: $\boxed{\text{True}}$

$B - A \subseteq B$

• Explanation: Similarly, the set $B - A$ contains all elements that are in set $B$ but not in set $A$. Every element in $B - A$ must be an element of $B$. Therefore, $B - A$ is a subset of $B$.
• Truth Value: $\boxed{\text{True}}$

If $A = B$, then $A + B = \phi$

• Explanation: Assuming $A+B$ denotes the symmetric difference $A \Delta B = (A \setminus B) \cup (B \setminus A)$. If $A = B$, then $A \setminus B = \phi$ (since all elements of $A$ are also in $B$) and $B \setminus A = \phi$ (since all elements of $B$ are also in $A$). Therefore, $A \Delta B = \phi \cup \phi = \phi$.
• Truth Value: $\boxed{\text{True}}$

$A - B \neq B - A$

• Explanation: This statement claims that $A - B$ is always not equal to $B - A$. This is not universally true. For example, if $A = \{1\}$ and $B = \{1\}$, then $A - B = \phi$ and $B - A = \phi$. In this case, $A - B = B - A$. Since we found a counterexample, the statement is false.
• Truth Value: $\boxed{\text{False}}$

$A - B = A - (A \cap B)$

• Explanation: The set $A - B$ contains elements in $A$ but not in $B$. The set $A - (A \cap B)$ contains elements in $A$ but not in the intersection of $A$ and $B$. An element $x$ is in $A$ and not in $B$ if and only if $x$ is in $A$ and $x$ is not in $(A \cap B)$. This is a fundamental identity in set theory.
• Truth Value: $\boxed{\text{True}}$

$A - \phi = A, A - A = \phi$

• Explanation: This statement consists of two parts:
  • $A - \phi = A$: The set $A - \phi$ contains elements in $A$ that are not in the empty set