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Cardinality Examples

Example 1: Disjoint Sets​

Let,
A = {1, 2, 3}
B = {4, 5, 6, 7}

Then:

  • n(A) = 3
  • n(B) = 4
  • A ∩ B = ∅, so n(A ∩ B) = 0
  • n(A ∪ B) = n(A) + n(B) = 3 + 4 = 7

Example 2: Intersecting Sets​

Let
A = {1, 2, 3, 4, 5, 6}
B = {4, 5, 6, 7}

Then:

  • A ∩ B = {4, 5, 6}, so n(A ∩ B) = 3
  • n(A) = 6
  • n(B) = 4
  • n(A ∪ B) = n(A) + n(B) − n(A ∩ B) = 6 + 4 − 3 = 7

Example 3: Real-Life Application​

Q1. In a village of 500 people:

  • 325 drink filtered water
  • 230 drink boiled water
  • Find how many drink both.
Show solution

Let,
n(F)=325n(F) = 325
n(B)=230n(B) = 230
n(F∪B)=500n(F ∪ B) = 500
Let, x=n(F∩B)x = n(F ∩ B)

Then:
n(F ∪ B) = n(F) + n(B) - n(F ∩ B)

⇒ 500 = 325 + 230 − x

⇒ x = 55

So, 55 people drink both filtered and boiled water.