Set Algebra & Laws — Set Theory

This is a continuation of my previous post about Set Theory. If you have not read that yet, start there.

There are few rules governing the universe of sets.

Algebra

Intersection

Intersection of two sets is a set that contains elements present in both sets.

Overlapping area

A ∩ B = {x | x ∈ A ∧ x ∈ B}

Union

Union of two sets is a set of elements that are present in at least one set.

A ∪ B = {x | x ∈ A ∨ x ∈ B}

Set Difference

Difference between two sets is a set of elements that is present in the first set but not in the second.

Green area

A \ B = {x | x ∈ A ∧ x ∉ B}

Symmetric difference

Symmetric difference between two sets is a set of elements that is present only in one set.

Green area

A ⊕ B = (A \ B) ∪ (B \ A)

Complement

Complement of a set is everything that is not found within that set but found within the universe of discourse.

Green area

A^c = X \ A

Please note that medium.com does not support superscript for letters. Consider that the letter c in A^c above is superscripted.

Laws of Sets

Absorption

Intersection or union of a set with itself results in the original set.

A ∩ A = A

A ∪ A = A

Commutativity

Changing operands does not change the result of an intersection or an union operation.

A ∩ B = B ∩ A

A ∪ B = B ∪ A

Associativity

Groups do not have a significance when intersecting sets or taking the union of sets

A ∩ (B ∩ C) = (A ∩ B) ∩ C

A ∪ (B ∪ C) = (A ∪ B) ∪ C

Distributivity

Algebraic distributive laws apply when intersecting sets or taking the union of sets

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

De Morgan’s laws

Complement of an intersection or an union is equivalent to the union or intersection of the complements.

(A ∩ B)^c = A^c ∪ B^c

(A ∪ B)^c = A^c ∩ B^c

Duality

Complement of the universe of discourse is the empty set while the complement of the empty set is the universe of discourse.

U^c = ∅

∅^c = U

Identity

Identity of a set is the result of a set union with the empty set or a set intersection with the universe of discourse.

A ∪ ∅ = A

A ∩ X = A

Dominance

The empty set results in a set intersection with an empty set while universe of discourse results in a union of any set with universe of discourse.

A ∩ ∅ = ∅

A ∪ X = X

Complementation

Set has no elements in common with its complement while the union of a set and it’s complement is everything.

A ∩ A^c = ∅

A ∪ A^c = X

Contraposition

  • If the complement of A is a subset of the complement of B, then B is a subset of A

A^c ⊆ B^c ≡ B ⊆ A

  • If A is a subset of the complement of B, then B is a subset of the complement of A

A ⊆ B^c ≡ B ⊆ A^c

  • If the complement of A is a subset of B, then the complement of B is a subset of A

A^c ⊆ B ≡ B^c ⊆ A


Originally published at medium.com on September 14, 2017.

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