134
Union of sets
Textbook: Part 4, lesson 134, pp. 75–77
GoalThe union of sets: A ∪ B, counting common elements once; the number of elements.
New words
union · birlashmathe sign ∪ · belgi ∪common element · umumiy elementonce · bir marta
Explanation
All the elements of the sets A and B together form their union, written A ∪ B. Common elements are written only once in the union. To find the number of elements in the union we add the numbers of elements and subtract the number of common elements once: n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
Worked examples
A = {1, 3, 5, 7}, B = {5, 7, 9, 11}. A ∪ B = {1, 3, 5, 7, 9, 11}, A ∩ B = {5, 7}. The number of elements: 4 + 4 − 2 = 6.
15 children go to the English club and 12 to the chess club, and 5 children go to both. In all, 15 + 12 − 5 = 22 children go to the clubs.
Class activity
“Two-hoop count”: two hoops are laid partly overlapping. Children stand according to two properties such as “likes reading” and “likes drawing”. The class counts all children in the hoops without counting a child in the overlap twice.
Practice
1
A = {a, b, c}, B = {c, d}. Write A ∪ B.
{a, b, c, d}
2
Set A has 6 elements, set B has 9 elements and A ∩ B has 4 elements. How many elements does A ∪ B have?
11
3
In a class 14 children go to swimming and 11 to football, and 6 go to both. If every child goes to at least one club, how many children are there?
19
4
A is the set of even natural numbers up to 10 and B the set of natural multiples of 3 up to 10. Write A ∪ B.
{2, 3, 4, 6, 8, 9, 10}
5
In the previous task A has 5 elements, B has 3 elements and A ∩ B = {6}. How many elements does A ∪ B have?
7