7.12: Introduction to set theory
- Page ID
- 192665
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)(1) | ⟦cat⟧ = the set of all cats in the actual world |
(2) | ⟦ Panks ⟧ = the individual Panks in the actual world |
![A photo of two Siberian Forest Cats on a couch, looking at the camera. The one in the foreground is grey and striped. The one in the background is brown and striped.](https://socialsci.libretexts.org/@api/deki/files/131186/pancakes-waffles-822x1024.jpg?revision=1)
(3) | ⟦ Panks is a cat ⟧ = T if the individual Panks is a member of the set of all cats in the actual world, F otherwise |
(4) | ⟦ Karti is a cat ⟧ = T if the individual Karti is a member of the set of all cats in the actual world, F otherwise |
![A photo of a corgi looking at the camera with his tongue out.](https://socialsci.libretexts.org/@api/deki/files/131187/karti1-300x300.jpg?revision=1)
![A diagram. On the left, a list of cats: p = Panks, w = Waffy, s = Sugar, l = Lilly, b = Brezel. To the right of the cat list is a list of dogs: k = Karti, h = Howard, i = Hildy, m = P-Max, a = Archie. To the right of the list of animals are two circles, one above another. Top circle is labeled "the set of CATS", and contains the letters p, w, s, b, l. The bottom circle is labeled "the set of DOGS, and contains the letters k, i, h, m, and a.](https://socialsci.libretexts.org/@api/deki/files/131188/sets1-1024x598.png?revision=1)
![The same image as Figure 7.19, but in the circle that represents the set of cats, there is a smaller circle around p, l, and w. This smaller circle is labeled "the set of SIBERIAN CATS".](https://socialsci.libretexts.org/@api/deki/files/131189/sets2-1024x597.png?revision=1)
(5) | A is a subset of B if all members of A are also members of B. |
(6) | A is a proper subset of B if all members of A are also members of B, and there is at least one member of B which is not a member of A. |
Table \(\PageIndex{1}\). Set theory notation.
Notation | Meaning |
A | The set A. Capital letters are used to refer to sets. |
{ a, b, c } | The set containing members a, b, and c. |
b ∈ A | b is a member of A. |
d ∉ A | d is not a member of A. |
A⊆B | A is a subset of B. |
B⊇A | B is a superset of A. |
A⊂B | A is a proper subset of B. |
B⊃A | B is a proper superset of A. |
A = B | A is identical to B (that is, A and B have exactly the same members) |
|A| | The cardinality of A. That is, the number of members in A. |
|A| = 3 | The cardinality of A is equal to 3. That is, there are three members in A. |
A∩B | The intersection of A and B. This set contains objects that are both a member of A and a member of B. |
A∪B | The union of A and B. This set contains objects that are either a member of A or a member of B. |
![A diagram for C∩G (intersection of C and G). Blue circle represents C. Pink circle represents G. The two partially overlap like a Venn diagram. The overlapping purple part is labeled C∩G.](https://socialsci.libretexts.org/@api/deki/files/131190/intersection-1024x435.png?revision=1)