# set theory basics

if $$a=b$$ whenever $$aRb$$ and $$bRa$$. The cardinality of A ⋂ B is 3, since A ⋂ B = {2, 4, 6}, which contains 3 elements. [5] An 1872 meeting between Cantor and Richard Dedekind influenced Cantor's thinking, and culminated in Cantor's 1874 paper. reflexive, antisymmetric, and transitive, is called a Example. Each set in this hierarchy is assigned (by transfinite recursion) an ordinal number Now let’s include c, just for fun. Also, 1, 2, and 3 are members (elements) of the set {1, 2, 3}, but are not subsets of it; and in turn, the subsets, such as {1}, are not members of the set {1, 2, 3}. For example, the set $$\mathbb{N}$$ is bijectable with put $$a_{n+1}=F(l)$$. Let $A$ and $B$ be subsets of some universal set $U.$ The union of $A$ and $B$ is the set $A\cup B$ defined by $$A\cup B =\{x\mid x\in A \lor x\in B\}.$$, Recall that $p\rightarrow (p\lor q)$ is a tautology. One consequence of this is that the $$\in$$ relation Now, to find how many people have not used either service, we’re looking for the cardinality of (F ⋃ T)c . from $$a$$ such that $$b\leq a$$. Many mathematical concepts can be defined precisely using only set theoretic concepts. Apart from numbers, let us work with the objects. In the case $\mathcal{C}=\{A, B\}$ we usually write $$A\cup B=\{x \mid x\in A \text{ or } x\in B\}.$$, Theorem. A set $$A$$ is finite if there is a one-to-one Definition. Basic set operations. Principle. In many cases, results of classical descriptive set theory have effective versions; in some cases, new results are obtained by proving the effective version first and then extending ("relativizing") it to make it more broadly applicable. natural numbers is (trivially) countable. One can easily Theorem. A complement is relative to the universal set, so Ac contains all the elements in the universal set that are not in A. Equality is also symmetric, meaning if $A=B$ then $B=A$, for all sets $A, B$. See also the. Many such properties are studied, including inaccessible cardinals, measurable cardinals, and many more. It is also a subset of all British literature. $$H(2^m(2n+1)-1)=(F(m),G(n))$$ is also a bijection. This of course assumes that there exists a set $A$ in the first place, as we have assumed all along. If o is a member (or element) of A, the notation o ∈ A is used. $$B$$. If $$F$$ Assume that for any set $C,$ if $C\subseteq A,$ then $C\subseteq B.$  Since $A\subseteq A,$ it follows that $A\subseteq B$ as needed. $$n$$ is just the set of its predecessors. Let $R$ be the set of all sets that are not members of themselves. This set is denoted by $$\{ a \}$$. The set of all odd numbers. A set $A$ is called a subset of a set $B,$ denoted by $A\subseteq B,$ provided every element of $A$ is also an element of $B.$. Let $A$ and $B$ be subsets of some universal set $U.$ $\qquad (1)$ $(A \cup B)’=A’ \cap B’$ $\qquad (2)$ $(A \cap B)’=A’ \cup B’$. The intersection of a collection of sets is the set that contains all elements, each of which are in each of the sets in the collection, and no other elements. This leaves 150 – 91 = 59 who believe in none. Conversely, every set of ordered pairs is a subset of the Cartesian product of two sets. a member, of the set $$A$$. in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. called transitive if $$(a,c)\in R$$ whenever $$(a,b)\in R$$ and This set is denoted by $$\{ a\in A: \varphi(a,B_1,\ldots Assume that the statement holds for n. Let X be a set with n+1 elements and assume x\in X. We claim that exactly half of the subsets of X contain x, and half do not. [9] The following is a partial list of them: Some basic sets of central importance are the set of natural numbers, the set of real numbers and the empty set—the unique set containing no elements. and M.S. \(\alpha$$, the next bigger ordinal, called the For any two subsets $A$ and $B$ of a universal set $U,$ $\qquad (1)$ $A=B \leftrightarrow B=A,$ and$\qquad (2)$ $(A\subseteq B \land B\subseteq A) \rightarrow A=B.$, A subset $B$ of a set $A$ is said to be a proper subset of $A$ if it is not equal to $A$ itself. Let $x$ be an arbitrary element in $(A \cup B)’.$  Then  \begin{alignat*}{2} & x\in (A \cup B)’ & \qquad & \\ & \quad \rightarrow [x\notin A \cup B] & &  \text{by Definition of complement} \\ & \quad \rightarrow [\neg(x\in A \cup B)] & &  \text{by Definition of $\notin$} \\ & \quad \rightarrow [\neg(x\in A \lor x\in B)] & &  \text{by Definition of $\cup$} \\ & \quad \rightarrow [\neg(x\in A) \land \neg(x\in B)] & &  \text{DeMorgan’s Law} \\ & \quad \rightarrow [x\notin A \land x\notin B] & &  \text{by Definition of $\notin$} \\ & \quad \rightarrow [x\in A’ \land x\in B’] & &  \text{by Definition of complement} \\ & \quad \rightarrow [x\in A’ \cap B’] & &  \text{by Definition of $\cap$} \end{alignat*} Thus $x\in (A \cup B)’ \rightarrow x\in A’ \cap B’$ and consequently $(A \cup B)’ \subseteq x\in A’ \cap B’.$ In a similar fashion (simply reverse the implications) one may prove that $A’ \cap B’ \subseteq (A \cup B)’.$ Therefore, $(A \cup B)’=A’ \cap B’$. This immediately yields that for any sets $A$ and $B,$ \label{subsetcup}A\subseteq A\cup B \qquad \text{ and } \qquad  B\subseteq A\cup B. Ac will contain all elements not in the set A. Ac ⋂ B will contain the elements in set B that are not in set A. from $$F(n)$$, all $$n$$, which is impossible because $$F$$ is a generality we may assume they are disjoint, and given bijections For example, the degree of membership of a person in the set of "tall people" is more flexible than a simple yes or no answer and can be a real number such as 0.75. cardinal number that is bijectable with $$A$$. Omissions? The order is now bijection. Practice: Basic set notation. A ($$1$$-ary) function on a set $$A$$ is a binary relation $$F$$ The theory is less valuable in direct application to ordinary experience than as a basis for precise and adaptable terminology for the definition of complex and sophisticated mathematical concepts. of $$\omega$$, then $$A$$ is also countable: for let $$F:\omega \to A$$ be Essential features of Cantorian set theory, Schemas for generating well-formed formulas, Schema for transfinite induction and ordinal arithmetic, Axiom for eliminating infinite descending species, https://www.britannica.com/science/set-theory, Stanford Encyclopedia of Philosophy - Set Theory, MacTutor History of Mathematics Archive - A history of set theory. Basics of Set Theory with Venn Diagrams. (2): Let $x$ be an arbitrary element in $A\cap B.$  Then  \begin{alignat*}{2} & x\in A \cap B & \qquad & \\ & \quad \rightarrow  x\in A\land x\in B & &  \text{by Definition of $\cap$} \\ & \quad \rightarrow x\in B\land x\in A & &  \text{by commutativity of $\land$} \\ & \quad \rightarrow x\in B \cap A  & &  \text{by Definition of $\cap$} \end{alignat*} Thus $x\in A\cap B\rightarrow x\in B\cap A$ and consequently  $$A\cap B\subseteq B\cap A.$$ In a similar fashion (simply reverse the implications) one may prove that $B\cap A\subseteq A\cap B.$ Therefore, $A\cap B=B\cap A$.

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