📐 Module 01 · Interactive
Set Operations with Venn Diagrams
— Interactive Venn Diagram —
👆 Click any region on the diagram to explore it
UNION
Union (A ∪ B)
A ∪ B = { x : x ∈ A or x ∈ B }
The union contains all elements from set A or set B (or both).
📌 n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
🟢 Shaded area = everything in U but outside circle A.
📐 A′ grows larger when A is smaller.
🔢 n(A′) = n(U) − n(A)
🟠 Shaded area = everything in U but outside circle B.
📐 B′ grows larger when B is smaller.
🔢 n(B′) = n(U) − n(B)
📝 Module 02 · Worked Examples
Worked Examples
Example 1 of 4
P = {multiples of 3 < 20}, Q = {multiples of 2 < 20}. Find P ∪ Q and P ∩ Q.
P={3,6,9,12,15,18}Q={2,4,6,8,10,12,14,16,18}
1
P ∪ Q = all unique elements = {2,3,4,6,8,9,10,12,14,15,16,18}
2
P ∩ Q = elements in both = {6,12,18}
Example 2 of 4
A = {a,b,c,d,e}, B = {c,d,e,f,g}. Find A ∪ B and A − B.
A={a,b,c,d,e}B={c,d,e,f,g}
1
A ∪ B = combine all unique = {a,b,c,d,e,f,g}
2
A − B = in A, not in B = {a,b}
Example 3 of 4
U = {1..10}, A = {2,4,6,8,10}. Find A′.
U={1..10}A={2,4,6,8,10}
1
A′ = U − A = elements of U NOT in A
2
Remove {2,4,6,8,10} → A′ = {1,3,5,7,9}
Example 4 of 4
In a class of 40: 25 play football, 15 play cricket, 10 play both. Find at-least-one and neither.
n(F)=25n(C)=15n(F∩C)=10Total=40
1
n(F∪C) = 25 + 15 − 10 = 30
2
Neither = 40 − 30 = 10
✏️ Module 03 · Practice
Practice Questions
📝 Written Solution
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