Meaning of union and intersection (3rd grade language, statistics meaning)
Consider two groups of students: A = students who like (or do) one thing, and B = students who like (or do) another thing.
- Intersection A ∩ B means both (the overlap).
- Union A ∪ B means either one or the other or both (everything inside the circles).
- Neither means not in A and not in B.
Key counting rule (addition rule):
When counting the union, the overlap gets counted twice if numbers are simply added, so the overlap must be subtracted once:
\[ |A \cup B| = |A| + |B| - |A \cap B| \]
Worked union and intersection question 1 (fruit preferences)
A class has 30 students. 18 like apples, 12 like bananas, and 7 like both apples and bananas.
Find:
- How many like apples or bananas? (the union A ∪ B)
- How many like both? (the intersection A ∩ B)
- How many like neither apples nor bananas?
Step 1: Identify the intersection
The problem states the overlap directly: \[ |A \cap B| = 7 \]
Step 2: Compute the union using the addition rule
\[ |A \cup B| = |A| + |B| - |A \cap B| \] \[ |A \cup B| = 18 + 12 - 7 = 23 \]
So, 23 students like apples or bananas (including those who like both).
Step 3: Compute “neither” using the total
Students who like neither are outside the union: \[ \text{Neither} = 30 - |A \cup B| = 30 - 23 = 7 \]
Step 4: (Optional check) Find “only apples” and “only bananas”
\[ \text{Only apples} = |A| - |A \cap B| = 18 - 7 = 11 \] \[ \text{Only bananas} = |B| - |A \cap B| = 12 - 7 = 5 \]
Check total: \[ 11 + 7 + 5 + 7 = 30 \]
| Quantity | Meaning | Value |
|---|---|---|
| |A| | Like apples | 18 |
| |B| | Like bananas | 12 |
| |A ∩ B| | Like both (intersection) | 7 |
| |A ∪ B| | Like apples or bananas (union) | 23 |
| Neither | Like neither apples nor bananas | 7 |
Worked union and intersection question 2 (after-school clubs)
In a grade, 25 students joined the Art club, 19 joined the Music club, and 10 joined both clubs. There are 40 students total.
Find how many joined Art or Music, and how many joined neither.
Step 1: Union (Art or Music)
\[ |A \cup B| = 25 + 19 - 10 = 34 \]
Step 2: Neither
\[ \text{Neither} = 40 - 34 = 6 \]
Worked union and intersection question 3 (pets at home)
A survey asks 28 students: “Do you have a dog?” and “Do you have a cat?” Results: 14 have a dog, 9 have a cat, and 4 have both.
Find how many have a dog or a cat, and how many have neither.
Step 1: Union (dog or cat)
\[ |A \cup B| = 14 + 9 - 4 = 19 \]
Step 2: Neither
\[ \text{Neither} = 28 - 19 = 9 \]
Common mistakes in union and intersection questions
- Double-counting the overlap: adding both groups without subtracting the intersection counts “both” twice.
- Mixing up “either” and “both”: union means “either or both,” intersection means “both only.”
- Forgetting the total: “neither” requires the total number of students or objects.
Quick summary (memorize-friendly)
\[ \text{Union} = \text{Group A} + \text{Group B} - \text{Both} \] \[ \text{Neither} = \text{Total} - \text{Union} \]