The Inclusion-Exclusion Principle for Ternary Relationships | Core Mathematics SHS 1 SEM 1 WEEK 3 (WASSCE & NaCCA Aligned)

Organizing Complex Real-World Data

The real number system provides the fundamental toolkit for organizing information, and when data categories begin to overlap—such as analyzing customer preferences for Kenkey, Banku, and Rice—we must move beyond simple two-set thinking. The three-set Venn diagram is our core analytical tool, allowing us to visually dissect a Universal Set (U) into eight distinct, non-overlapping partitions.

Consider a survey conducted in a typical Ghanaian market regarding three staple foods: Rice (R), Banku (B), and Kenkey (K). If we merely asked how many people like each food individually, we would significantly overcount the total population because individuals often belong to multiple categories. The mathematical power of set analysis lies in its ability to isolate those overlaps and subtract the redundancy.

• The universal set (U) represents everyone in the survey, such as all students in a class.
• The circles R, B, and K represent those who possess the specific characteristic (e.g., those who like each food).

Mastering the Geometry of Three Sets

A three-set Venn diagram is divided into eight mutually exclusive regions. Understanding the precise notation for each region is crucial for solving application problems. These eight regions account for every possible combination within the universal set.

• **The Center Region ($R \cap B \cap K$):** This represents the elements that belong to *all three* sets. In our market analogy, these are the people who enjoy Rice, Banku, *and* Kenkey. This region is the algebraic anchor for all three-set problems.
• **The Two-Set Intersections (e.g., $R \cap B \cap K’$):** These three regions represent those elements that belong to *exactly two* sets. For instance, $R \cap B \cap K’$ describes those who like Rice and Banku, but specifically *not* Kenkey. These regions must be isolated from the full intersection $n(R \cap B)$, which includes the center region.
• **The ‘Only’ Regions (e.g., $R \cap B’ \cap K’$):** These three regions represent those elements that belong to *exactly one* set. $R \cap B’ \cap K’$ are the people who like Rice and nothing else among the surveyed items.
• **The Complement ($(R \cup B \cup K)’$):** This final region is everything outside the three circles, representing the elements in the universal set that belong to *none* of the three categories (e.g., people who only eat Fufu).

Visualizing these regions by shading them in your notebook—starting with the central intersection and working outwards—is the most effective pedagogical strategy for internalizing set relationships.

Formalizing Relationships: Notation and Logic

We use sophisticated notation to formally define these regions and relationships. One of the most powerful tools for manipulating set logic is **De Morgan’s Law**.

De Morgan’s Law relates the complement of unions and intersections:

• $$(A \cap B)’ = A’ \cup B’$$
• $$(A \cup B)’ = A’ \cap B’$$

This law shows that excluding the intersection of A and B is the same as including everything that is outside of A *or* outside of B. Conversely, excluding the union of A and B is the same as only including elements that are simultaneously outside of A *and* outside of B. While most introductory problems focus on two sets for verification (using shading exercises), its principles extend fundamentally to three-set logic, helping us define complex complements like $(A \cup B \cup C)’$.

Furthermore, translating everyday language into set notation is the bridge between the real world and mathematical analysis. For example, “students who speak Twi and Ga but not Ewe” translates precisely to $T \cap G \cap E’$, provided T, G, and E represent the respective language speakers.

The Inclusion-Exclusion Formula: Calculating Totals

When solving word problems—especially those where categories overlap—we use the Principle of Inclusion-Exclusion to find the total number of elements in the union of the three sets, $n(A \cup B \cup C)$.

We must develop a systematic strategy, often called the **”Fill from the Middle”** strategy, to construct the Venn diagram accurately. We start by placing the known value for the intersection of all three, $n(A \cap B \cap C)$, in the innermost region. We then use this value to calculate the exact count for the regions belonging to *only* two sets.

The final equation for the total number of elements in the union is:

$$\mathbf{n(A \cup B \cup C) = n(A) + n(B) + n(C)}$$
$$\text{(Include all single counts)}$$

$$\mathbf{- [n(A \cap B) + n(A \cap C) + n(B \cap C)]}$$
$$\text{(Exclude the double counts because they were counted twice)}$$

$$\mathbf{+ n(A \cap B \cap C)}$$
$$\text{(Include the triple count back because it was added three times and subtracted three times)}$$

This systematic addition and subtraction ensures that every single element that belongs to at least one set is counted exactly once. This formula is vital for real-world applications, such as analyzing census data or market research results. Mastering the setup of the Venn diagram using the “Fill from the Middle” strategy is generally more reliable for solving specific counts (like “only Rice”) than relying solely on algebraic manipulation of the full formula, especially when several variables are unknown. The visual representation ensures every region is accounted for before solving for the unknown variable, often denoted as $x$.

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