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Core Concept

The union of two sets A and B, written as A ∪ B, is the set containing all elements that belong to A, or to B, or to both.

Mathematical Definition

$A \cup B = \{x | x \in A \text{ or } x \in B\}$

An element belongs to the union if it belongs to at least one of the sets.

Visual Representation

In a Venn diagram, the union is the entire region covered by both circles.

    A         B
  (###[###]###)

The entire shaded region represents A ∪ B.

Key Properties

1. Commutative Property

$A \cup B = B \cup A$

2. Associative Property

$(A \cup B) \cup C = A \cup (B \cup C)$

3. Identity Property

$A \cup \emptyset = A$

4. Idempotent Property

$A \cup A = A$

5. Union with Universal Set

$A \cup U = U$ (where U is the universal set)

6. Distributive Property

$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$

7. De Morgan's Laws

$\overline{A \cup B} = \overline{A} \cap \overline{B}$

Worked Examples

Example 1: Finite Sets

Given: A = {1, 2, 3}, B = {3, 4, 5}

Find: A ∪ B

Solution: All elements from both sets (no duplicates)

Answer: A ∪ B = {1, 2, 3, 4, 5}

Example 2: Interval Union

Given: A = [1, 4], B = [3, 7]

Find: A ∪ B

Solution: The combined coverage is [1, 7]

Answer: A ∪ B = [1, 7]

Example 3: Disjoint Intervals

Given: A = [1, 2], B = [4, 5]

Find: A ∪ B

Solution: No overlap, union is both intervals

Answer: A ∪ B = [1, 2] ∪ [4, 5] (cannot be simplified)

Example 4: Set-Builder Notation

Given: A = {x | x ≤ 3}, B = {x | x ≥ 1}

Find: A ∪ B

Solution: Elements ≤ 3 OR ≥ 1 covers all real numbers

Answer: A ∪ B = ℝ (all real numbers)

CSCA Practice Problems

💡 Note: The following practice problems are designed based on the CSCA exam syllabus.

Example 1: Basic (Difficulty ★☆☆☆☆)

If A = {1, 3, 5} and B = {2, 3, 4}, find A ∪ B.

Options:

• A. {3}
• B. {1, 2, 3, 4, 5}
• C. {1, 5}
• D. {2, 4}

Solution: Combine all elements: 1, 2, 3, 4, 5

Answer: B

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Example 2: Intermediate (Difficulty ★★★☆☆)

Given A = {x | -1 ≤ x < 3} and B = {x | 2 < x ≤ 5}, find A ∪ B.

Solution:

A = [-1, 3) and B = (2, 5]

Since 2 < 3, the intervals overlap: $A \cup B = [-1, 3) \cup (2, 5] = [-1, 5]$

Answer: [-1, 5]

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Example 3: Advanced (Difficulty ★★★★☆)

If A ∪ B = A, what is the relationship between sets A and B?

Solution:

If A ∪ B = A, then all elements of B must already be in A.

This means B ⊆ A (B is a subset of A).

Answer: B ⊆ A

Relationship with Intersection

Counting Formula

For finite sets: $|A \cup B| = |A| + |B| - |A \cap B|$

Example: If |A| = 10, |B| = 8, |A ∩ B| = 3, then: $|A \cup B| = 10 + 8 - 3 = 15$

De Morgan's Laws

$\overline{A \cup B} = \overline{A} \cap \overline{B}$ $\overline{A \cap B} = \overline{A} \cup \overline{B}$

Common Mistakes

❌ Mistake 1: Counting Elements Twice

Wrong: If A = {1, 2, 3} and B = {2, 3, 4}, then |A ∪ B| = 6 ✗

Correct: |A ∪ B| = 3 + 3 - 2 = 4 ✓

❌ Mistake 2: Confusing Union with Intersection

Wrong: A ∪ B includes only common elements ✗

Correct: A ∪ B includes ALL elements from both sets ✓

❌ Mistake 3: Simplifying Disjoint Intervals

Wrong: [1, 2] ∪ [4, 5] = [1, 5] ✗

Correct: [1, 2] ∪ [4, 5] cannot be simplified (gap at (2, 4)) ✓

Study Tips

1. ✅ "OR" thinking: Union means OR - element satisfies EITHER condition
2. ✅ Use counting formula: For finite sets, remember to subtract the intersection
3. ✅ Check for gaps: Interval unions may not simplify if there's a gap
4. ✅ Master De Morgan's Laws: Essential for complement problems

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💡 Exam Tip: Union problems often combine with counting. Always use |A ∪ B| = |A| + |B| - |A ∩ B| for finite sets!