3
Union and intersection of two sets with Venn diagrams
Mathematics Prep 1 Term 1, Unit 1, Lesson 3
GoalFind the union and the intersection of two sets and show them in a Venn diagram.
New words
union A ∪ B · all elements that are in A or in B (or both)intersection A ∩ B · the elements that are in both A and Bdisjoint sets · sets with no common element: A ∩ B = ∅Venn diagram · circles that show how sets overlap
Explanation
A Venn diagram draws each set as a circle. The overlap holds the elements in both sets. The union A ∪ B collects everything in either circle, and a common element is written only once. The intersection A ∩ B is just the overlap. If one set is inside the other, say C ⊂ D, then C ∪ D = D and C ∩ D = C. If two sets share nothing they are disjoint and their intersection is ∅. For any set A, A ∪ ∅ = A and A ∩ ∅ = ∅.
★Remember
A ∪ B: in A or B, common ones once. A ∩ B: in both A and B.
Worked examples
A = {2, 4, 6, 8, 10} (even numbers up to 10) and B = {3, 6, 9} (multiples of 3 up to 10). The common element is 6, so A ∩ B = {6}. The union lists 6 once: A ∪ B = {2, 3, 4, 6, 8, 9, 10}.
One set inside another. C = {1, 2} and D = {1, 2, 3, 4}. Every element of C is in D, so C ⊂ D. Then C ∪ D = {1, 2, 3, 4} = D and C ∩ D = {1, 2} = C.
Disjoint sets. E = {1, 3, 5} and F = {2, 4, 6} have no common element, so E ∩ F = ∅. The union is E ∪ F = {1, 2, 3, 4, 5, 6}. The two circles do not touch.
Class activity
Two baskets. With a grown-up, put some fruit names on paper slips: basket A holds apple, banana, orange and basket B holds banana, grape. Make a Venn diagram with two loops of string. What is in A ∩ B and A ∪ B?
Practice
1
P = {1, 2, 3, 4} and Q = {3, 4, 5, 6}. Find P ∪ Q and P ∩ Q.
P ∪ Q = {1, 2, 3, 4, 5, 6}; P ∩ Q = {3, 4}.
2
A is the set of multiples of 2 from 1 to 12 and B the multiples of 3 from 1 to 12. Find A ∩ B and A ∪ B.
A = {2, 4, 6, 8, 10, 12}, B = {3, 6, 9, 12}. A ∩ B = {6, 12}. A ∪ B = {2, 3, 4, 6, 8, 9, 10, 12}.
3
S = {1, 2} and T = {1, 2, 3, 5}. Find S ∩ T and S ∪ T. What is the link between S and T?
S ∩ T = {1, 2} = S and S ∪ T = {1, 2, 3, 5} = T. S is a subset of T.
4
M = {10, 20} and N = {30, 40}. Find M ∩ N and M ∪ N.
M ∩ N = ∅ (disjoint). M ∪ N = {10, 20, 30, 40}.
5
Mina writes {1, 2, 3} ∪ {3, 4} = {1, 2, 3, 3, 4}. What is wrong?
A common element is written once. The union is {1, 2, 3, 4}.