3.2: 4.2 The Boolean Algebra of Sets
- Page ID
- 9680
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)It is clear that set theory is closely related to logic. The intersection and union of sets can be defined in terms of the logical ‘and’ and logical ‘or’ operators. The notation {x | P(x)} makes it possible to use predicates to specify sets. And if A is any set, then the formula x ∈ A defines a one place predicate that is true for an entity x if and only if x is a member of A. So it should not be a surprise that many of the rules of logic have analogues in set theory.
For example, we have already noted that ∪ and ∩ are commutative operations. This fact can be verified using the rules of logic. Let A and B be sets. According to the definition of equality of sets, we can show that A∪B = B∪A by showing that ∀x((x ∈A ∪ B) ↔ (x ∈ B ∪ A)). But for any x,
x ∈ A ∪ B ↔ x ∈ A ∨ x ∈ B (definition of ∪)
↔ x ∈ B ∨ x ∈ A (commutativity of ∨)
↔ x ∈ B ∪ A (definition of ∪)
The commutativity of ∩ follows in the same way from the definition of ∩ in terms of ∧and the commutativity of ∧, and a similar argument shows that union and intersection are associative operations.
The distributive laws for propositional logic give rise to two similar rules in set theory. Let A, B, and C be any sets. Then
A∪(B∩C) = (A∪B)∩(A∪C)
and
A∩(B∪C) = (A∩B)∪(A∩C)
These rules are called the distributive laws for set theory. To verify the first of these laws, we just have to note that for any x,
x ∈ A∪(B∩C)
↔ (x ∈ A) ∨ ((x ∈ B) ∧ (x ∈ C)) (definition of ∪, ∩)
↔ ((x ∈ A) ∨ (x ∈ B)) ∧ ((x ∈ A) ∨ (x ∈ C)) (distributivity of ∨)
↔ (x ∈ A ∪ B) ∧ (x ∈ A ∪ C) (definition of ∪)
↔ x ∈ ((A ∪ B) ∩ (A ∪ C)) (definition of ∩)
The second distributive law for sets follows in exactly the same way.