الأسبوع 2 · الفترات: المحدودة وغير المحدودة والعمليات عليها: الاتحاد والتقاطع والفرق والمكملة · 3 دروس
3
Union, intersection, difference and complement of intervals
Mathematics Prep 2 Term 1, Unit 1, Lesson 2
GoalFind X ∪ Y, X ∩ Y, X − Y and the complement of intervals, checking every end point.
New words
union X ∪ Y · everything that is in X or in Y (or both)intersection X ∩ Y · only what is in both X and Ydifference X − Y · what is in X but not in Ycomplement X′ · R − X: everything in R that is not in X
Explanation
Intervals are sets, so we can combine them as we combine sets. Draw both on number lines, one above the other, then look at each part. For X = [−1, 4[ and Y = ]2, 6]: the union X ∪ Y is [−1, 6] because together they cover everything from −1 to 6. The intersection X ∩ Y = ]2, 4[ is where both are drawn. The difference X − Y = [−1, 2] keeps what is in X but not in Y; 2 stays because 2 is in X but not in Y. The complement of X in R is X′ = R − X = ]−∞, −1[ ∪ [4, ∞[. Always test the end points one by one. And the same operations work with other sets: Z ∩ ]−2, 3] = {−1, 0, 1, 2, 3}.
X = [−1, 4[ , Y = ]2, 6]
Result
X ∪ Y
[−1, 6]
X ∩ Y
]2, 4[
X − Y
[−1, 2]
Y − X
[4, 6]
X′
]−∞, −1[ ∪ [4, ∞[
Worked examples
Find [1, 4] ∪ ]3, 8[. The first covers 1 to 4 with both ends; the second covers 3 to 8 without its ends. Together they cover everything from 1 up to 8, and 8 is not included: [1, 8[.
Find ]1, 5] ∩ ]−∞, 3]. Numbers must be above 1 and also at most 3, so the answer is ]1, 3]. Check the ends: 1 is out (open in the first), 3 is in (closed in the second). Now find [0, 6[ − ]2, 4]: we remove 2 < x ≤ 4 and keep [0, 2] ∪ ]4, 6[.
Ages 25 to 45 are allowed in group A: [25, 45]. Group B takes ages 40 to 60: [40, 60]. Ages allowed in both groups: [25, 45] ∩ [40, 60] = [40, 45]. With integers: Z ∩ ]−2, 3] = {−1, 0, 1, 2, 3}; and N ∩ [−3, 0[ = ∅, because N has no negative numbers and 0 is not in [−3, 0[.
Class activity
Two clubs. Club A meets on days 1 to 20 of a month and club B on days 15 to 31. Write both as intervals, then find the days when both meet and the days when at least one meets. Check that the union has no gap.
