Algebra of Events

Let A and B be two events. Then

1. The event "either A or B" is said to occur if at least one of A and B occurs. Event
'A or B' does not occur if none of A and B occurs. We write it as 'A or B' also and represent it by A ∪ B or A + B.

2. The event "both A and B" is said to occur if both A as well as B occur simultaneously. Event 'A and B' does not occur if at least one of A and B does not occur. We write it as 'A and B' also and represent it by A ∩ B or AB.

3. The event "A not" occurs if A does not occur. It does not occur if A occurs. We write it as
A′, AC or S − A also.

4. The event "A but not B" is said to occur if A occurs but B does not occur. It does not occur if either A does not or A ∩ B occurs. It is represented by A− B or A∩Bc.

Some useful Results

If A′ = S − A, then

(1) (A ∪ B) ′ = A′ ∩ B′ (2) (A ∩ B)′ = A′ ∪ B′ (3) A − B = A   (A ∩ B) = A ∩ B′

(4) A′ − B = A′ ∩ B′ = (A ∪ B)′      (5) A′ − B′ = A′ ∩ B = B − A

(6) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)   (7) If A ⊂ B ∪ C ∪ D, then A = A ∩ (B ∪ C ∪ D)