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COMPLEMENT of set A in the universal set Ω is \(A^c\) = {x| x ∉ A}
INTERSECTION of sets A and B, A∩B: A∩B= {x| x∈A and x∈B}
The COMBINATION of sets A and B, A∪B: A∪B = {x| x∈A or x∈B}
Two sets are called MUTUAL SEPARATE (disjoint) if both are sliced is an empty set, meaning neither of them has the same elements/elements/members.
EMPTY SET is a set which has no members is called, denoted ∅
Properties of Set Processing
The Characteristic of Identicality to the Combination of and Slice
a. A ∪ ∅ = A
b. A ∪ Ω = Ω
c. A ∩ ∅ = ∅
d. A ∩ Ω = A
2. Similarity Properties of Combinations and Slice
a. A ∪ A = A
b. A ∩ A = A
3. Complementary Properties
a. A ∪ \(A^c\)= Ω
b. A ∩ \(A^c\)=∅
c. \((A^c)^c\) = A
4. Commutative Properties of Combinations and Slice
a. A ∪ B = B ∪ A
b. A ∩ B = B ∩ A
5. Association Properties of Combinations and Intersection of Sets
a. A ∪ (B ∪ C) = (A ∪ B) ∪ C
b. A ∩ (B ∩ C) = (A ∩ B) ∩ C
6. Distributive Property of Combination and Intersection
a. A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
b. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
c. (B ∩ C) ∪ A = (B ∪ A) ∩ (C ∪ A)
d. (B ∪ C) ∩ A = (B ∩ A) ∪ (C ∩ A)
7. De Morgan’s Law
a. \((A \cup B)^c\) = \(A^c\) ∩ \(B^c\)
b. \((A \cap B)^c\) = \(A^c\) ∪ \(B^c\)
Algebraic Operations on Events
Illustration:
Consider Illustration-1 on previous article
o C= {BBB}
o D={BBB, BBS, BSB, SBB}
o E={BBS, BSB, SBB}
Then:
o \(C \cup D\) = {BBB, BBS, BSB, SBB}
o \(D \cap E\) ={BBS, BSB, SBB}
o \(C \cap E\) = ∅ (C and E separate events)
o \(D^c\) ={ BSS, SBS, SSB, SSS}
