Union and Intersection of Intervals

Symbols for Set Operations

• Union symbol: ∪ — looks like a big capital U with no tail.

• Intersection symbol: ∩ — the upside-down U.

• Purpose:

  • Union (A ∪ B): combines all elements that are in A or in B or in both.

  • Intersection (A ∩ B): contains only the elements that are in both A and B.

• Notation reminder:

  • Union and intersection are the primary ways to “combine” sets when talking about ranges on the number line.

Union of Intervals: Example

• Given two open intervals: A = (2, 5) and B = (-3, 4).

  • Endpoints are not included in either interval (open intervals).

• Visual on the number line (conceptual):

  • A (in red) covers the region 2 < x < 5.

  • B covers -3 < x < 4.

• What does the union do?

  • It includes everything that lies in A or in B (or both).

• Overlap observation:

  • A and B overlap on the region (2, 4).

• Result of the union:

  • The union stretches from the leftmost left endpoint to the rightmost right endpoint, with endpoints not included because neither original interval includes them.

  • Therefore, A ∪ B = (-3, 5).

• Key takeaway:

  • When two intervals overlap, the union can be a single interval whose endpoints are the outermost bounds of the two intervals.

Union: General Rules (A = (a,b), B = (c,d))

• If the intervals overlap (or touch) so that ext{max}(a,c) < ext{min}(b,d), then the union is a single interval:

  • A \cup B = (\min(a,c), \max(b,d)).

• If there is a gap between A and B, the union is two separate intervals:

  • A \cup B = (a,b) \cup (c,d)\, (a,b)\cup(c,d) (two pieces).

• Important: these formulas assume open endpoints; for closed endpoints, inequalities change accordingly.

Intersection of Intervals: Example

• Given two open intervals: A = (-4, 1) and B = (-2, 8).

  • Endpoints are not included in either interval.

• Overlap analysis:

  • Left edge of overlap is max(-4, -2) = -2 (not included).

  • Right edge of overlap is min(1, 8) = 1 (not included).

• Result of the intersection:

  • A ∩ B = (-2, 1).

• Why this works:

  • Intersection contains exactly the points that lie in both A and B.

• General rule:

  • If \text{max}(a,c) < \text{min}(b,d), then the intersection is a nonempty interval:

  • A \cap B = (\max(a,c), \min(b,d)).

  • If the inequality is not satisfied, the intersection is empty: A \cap B = \emptyset.

Intersection: General Formula (A = (a,b), B = (c,d))

• Nonempty intersection condition:

  • \text{max}(a,c) < \text{min}(b,d).

• If nonempty:

  • A \cap B = (\max(a,c), \min(b,d)).

• If empty:

  • A \cap B = \emptyset.

Visual Interpretation on the Number Line

• Union:

  • Draw A and B, then shade everything that belongs to either interval.

  • If there is overlap, the shaded region forms a single continuous interval from the smallest left endpoint to the largest right endpoint.

• Intersection:

  • Shade only the region where A and B overlap.

  • The resulting interval is bounded by the larger of the two left endpoints and the smaller of the two right endpoints.

Practical Implications and Real-World Relevance

• Scheduling and time slots:

  • Union helps determine all time slots that are covered by any single schedule.

  • Intersection helps identify times when multiple schedules overlap (conflicts).

• Data ranges in measurements:

  • Union merges ranges of values observed; intersection identifies common value ranges across datasets.

• Risk assessment:

  • Union could represent total exposure range when combining multiple scenarios.

  • Intersection highlights conservative estimates where all scenarios agree.

• Problem-solving intuition:

  • Endpoints matter: open vs closed endpoints change whether a boundary value is included.

  • When intervals overlap, you often get a single interval for the union; when they don’t, you get multiple pieces.

Common Pitfalls to Avoid

• Assuming union always yields a single interval; it can be two or more disjoint intervals if there is a gap.

• Confusing endpoints: open intervals do not include endpoints; closed intervals do.

• Misidentifying the governing endpoints: for union, use the outermost endpoints if there is overlap; for intersection, use the inner endpoints.

• Forgetting the nonempty condition for intersection: if max(a,c) ≥ min(b,d), the intersection is empty.

Quick Reference Formulas (A = (a,b), B = (c,d))

• Union (overlapping case): A \cup B = (\min(a,c), \max(b,d)) if \max(a,c) < \min(b,d).

• Union (disjoint case): A \cup B = (a,b) \cup (c,d) if the intervals do not overlap.

• Intersection (nonempty): A \cap B = (\max(a,c), \min(b,d)) if \max(a,c) < \min(b,d).

• Intersection (empty): A \cap B = \emptyset if \max(a,c) \ge \min(b,d).

Practice Prompt (apply the concepts)

• Given A = (1, 3) and B = (2, 4):

  • A ∪ B = (1, 4) , since they overlap.

  • A ∩ B = (2, 3) , since the overlap region is from 2 to 3.

• Given C = (0, 1) and D = (1, 2):

  • A and B touch at x = 1 but do not overlap (open endpoints).

  • C ∪ D = (0,1) \cup (1,2) (two disjoint intervals).

  • C ∩ D = \emptyset .