Let's analyze each statement carefully and determine which one is not true:

1. Let $A \subseteq B$. Then $B' \subseteq A'$:
   This statement is true. The complement $B'$ (elements not in $B$) will include $A'$ (elements not in $A$), as $A \subseteq B$.

2. Let $A \subseteq B$. Then the union of $A$ and $B$ is precisely $B$:
   This statement is true. If $A \subseteq B$, combining $A$ and $B$ in a union doesn't add anything new, so the union is just $B$.

3. Let $A \subseteq B$. Then the intersection of $A$ and $B$ is precisely $A$:
   This statement is true. Since $A \subseteq B$, all elements of $A$ are already in $B$. Thus, the intersection of $A$ and $B$ is $A$.

4. Let $A \subseteq B$. Then the union of $A$ and $(B - A)$ is precisely $A$:
   This statement is not true.
   $B - A$ consists of elements in $B$ but not in $A$. When you take the union of $A$ and $(B - A)$, the result is $B$, not $A$.

Correct Answer:

The incorrect statement is the fourth one:
"Let $A \subseteq B$. Then the union of $A$ and $(B - A)$ is precisely $A$."

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Do you want detailed proofs or visualizations for any of these?
Here are five related questions for practice:

1. What is the difference between $B - A$ and $A'$ in set theory?
2. If $A \subseteq B$, what is the result of $(A \cap B)'$?
3. Can the union of two disjoint subsets ever equal one of the subsets?
4. Why does $(B - A) \cup A = B$ when $A \subseteq B$?
5. How can Venn diagrams help visualize these relationships?

Tip: When working with sets, always test statements with simple, concrete examples to confirm their validity.