Set operations — union, intersection, complement, difference with Venn diagrams

Question

If $A = \{1, 2, 3, 4, 5\}$ , $B = \{3, 4, 5, 6, 7\}$ , and the universal set $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ , find $A \cup B$ , $A \cap B$ , $A - B$ , $B - A$ , $A'$ , and verify $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ .

(CBSE 11 & JEE Main — sets chapter)

Solution — Step by Step

$A \cup B$ = all elements in A or B (or both): $\{1, 2, 3, 4, 5, 6, 7\}$

$A \cap B$ = elements in both A and B: $\{3, 4, 5\}$

$A - B$ = elements in A but not in B: $\{1, 2\}$

$B - A$ = elements in B but not in A: $\{6, 7\}$

$A' = U - A$ = elements in U but not in A: $\{6, 7, 8, 9, 10\}$

$n(A) = 5$ , $n(B) = 5$ , $n(A \cap B) = 3$ , $n(A \cup B) = 7$

n(A) + n(B) - n(A \cap B) = 5 + 5 - 3 = 7 = n(A \cup B) \checkmark

Why This Works

Set operations correspond to logical operations: union = OR, intersection = AND, complement = NOT. The formula $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ corrects for double-counting — elements in both sets get counted twice when we add $n(A)$ and $n(B)$ , so we subtract the intersection once.

graph TD
    A["Set Operation Problem"] --> B{"What's asked?"}
    B -->|"Elements in A OR B"| C["Union: A ∪ B<br/>Combine all elements"]
    B -->|"Elements in A AND B"| D["Intersection: A ∩ B<br/>Common elements only"]
    B -->|"Elements in A but NOT B"| E["Difference: A - B"]
    B -->|"Elements NOT in A"| F["Complement: A'<br/>= U - A"]
    B -->|"Count of union"| G["n(A∪B) = n(A) + n(B)<br/>- n(A∩B)"]
    G --> H{"Three sets?"}
    H -->|"Yes"| I["Add/subtract using<br/>inclusion-exclusion principle"]

Alternative Method — Venn Diagram Approach

Draw two overlapping circles inside a rectangle (universal set). Fill in the intersection first ( $\{3, 4, 5\}$ ), then the remaining parts of each circle ( $\{1, 2\}$ for A-only, $\{6, 7\}$ for B-only), then the outside ( $\{8, 9, 10\}$ ). Read any operation directly from the diagram.

For JEE word problems: when the question says ” $n$ students play cricket, $m$ play football, $k$ play both,” directly apply $n(A \cup B) = n + m - k$ . For three activities, use inclusion-exclusion: $n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)$ .

Common Mistake

Students write $A - B = B - A$ . These are NOT equal (set difference is not commutative). In our example, $A - B = \{1, 2\}$ while $B - A = \{6, 7\}$ . The symmetric difference $A \Delta B = (A - B) \cup (B - A) = \{1, 2, 6, 7\}$ is the operation that IS commutative. CBSE boards test this distinction explicitly.

Question

Solution — Step by Step

Why This Works

Alternative Method — Venn Diagram Approach

Common Mistake

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