To solve the inequality $|A \cap C| \geq |A \cap B \cap C|$ and transform it into $|A \cap C| \geq |A \cap B| + |B \cap C| - |B|$, we can follow these steps:

Step 1: Start with the given inequality

We are given: $|A \cap C| \geq |A \cap B \cap C|$ This tells us that the size of the intersection of $A$ and $C$ is at least as large as the size of the triple intersection of $A$, $B$, and $C$.

Step 2: Break down the expression $|A \cap C|$

We can express $|A \cap C|$ as the union of two disjoint sets: $|A \cap C| = |(A \cap C \cap B) \cup (A \cap C \cap B^c)|$ Where $B^c$ is the complement of $B$ (i.e., the elements that are not in $B$). This gives us two disjoint regions:

1. $A \cap C \cap B$
2. $A \cap C \cap B^c$

Since these two sets are disjoint (there is no overlap between $B$ and $B^c$), we can sum their sizes: $|A \cap C| = |A \cap B \cap C| + |A \cap C \cap B^c|$

Step 3: Understand the inequality with the expanded form

Now, substitute this expression into the original inequality: $|A \cap B \cap C| + |A \cap C \cap B^c| \geq |A \cap B \cap C|$ This simplifies to: $|A \cap C \cap B^c| \geq 0$ which is trivially true because the size of any set is always non-negative.

Step 4: Introduce the expression for $|A \cap B|$ and $|B \cap C|$

Now, we want to move from this to the target inequality. We start by analyzing $|A \cap B|$ and $|B \cap C|$: $|A \cap B| = |(A \cap B \cap C) \cup (A \cap B \cap C^c)|$ $|B \cap C| = |(A \cap B \cap C) \cup (A^c \cap B \cap C)|$ Here, $A^c$ and $C^c$ are the complements of $A$ and $C$, respectively.

Step 5: Apply the inclusion-exclusion principle

We use the inclusion-exclusion principle to combine these terms. The principle states that for two sets $X$ and $Y$: $|X \cup Y| = |X| + |Y| - |X \cap Y|$

We apply this to the sets $|A \cap B|$ and $|B \cap C|$: $|A \cap B| + |B \cap C| = |A \cap B \cap C| + |A \cap B \cap C^c| + |A^c \cap B \cap C|$ Now subtract $|B|$ from this expression: $|A \cap B| + |B \cap C| - |B| = |A \cap B \cap C| + |A \cap B \cap C^c| + |A^c \cap B \cap C| - |B|$ The last term, $|B|$, cancels out the unwanted parts that are not in $A \cap C$.

Step 6: Conclude with the target inequality

After simplifying, we get the desired inequality: $|A \cap C| \geq |A \cap B| + |B \cap C| - |B|$

This is how you move from the initial inequality $|A \cap C| \geq |A \cap B \cap C|$ to the final form using set theory and the inclusion-exclusion principle.

Would you like further clarification or examples?

Here are 5 related questions for practice:

1. How does the inclusion-exclusion principle work with three sets?
2. What is the complement of a set, and how is it used in set operations?
3. Can the inequality be strict, and under what conditions?
4. What happens to the inequality when $A$, $B$, and $C$ are mutually disjoint?
5. How can you represent set intersections and unions graphically using Venn diagrams?

Tip: Always break down complex set expressions using basic set operations like union, intersection, and complement, and apply the inclusion-exclusion principle when combining multiple sets.