Step By Step Calculus » 2.4 - Set Operations

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Synopsis
  • The Cartesian product A\times B A\times B of two sets A A and B B is the set of all possible ordered pairs whose first component is a member of A and whose second component is a member of B, i.e.
    \displaystyle A\times B = \{(x,y) | x\in A\;\mathrm{and}\;y\in B\}.
    \displaystyle A\times B = \{(x,y) | x\in A\;\mathrm{and}\;y\in B\}.
  • Operations and Properties of Sets: We introduce the idea of a universe of discourse\;U\;U from which we will choose the members of our sets. Common examples of U U are: \mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}, and \mathbb{R}^2 \mathbb{R}^2. Given two sets A A and B B from a universe U U, some operators and properties are:
    union: A\cup B=\left\{ x\in U: x\in A\text{ or }x\in B\right\} A\cup B=\left\{ x\in U: x\in A\text{ or }x\in B\right\} .
    intersection: A\cap B=\left\{ x\in U : x\in A\text{ and }x\in B\right\} A\cap B=\left\{ x\in U : x\in A\text{ and }x\in B\right\} . If two sets have no elements in common, the intersection is the empty set \emptyset \emptyset and we say that the sets are disjoint.
    complement: A^{C}\equiv A^{\prime }=\{x\in U : x\notin A\} A^{C}\equiv A^{\prime }=\{x\in U : x\notin A\}.
    proper subset: A\subset B A\subset B means that x\in A x\in A implies x\in B x\in B , and A \neq B A \neq B.
    subset: A\subseteq B A\subseteq B means A\subset B A\subset B or the two sets are equal.
    superset: B\supseteq A B\supseteq A means that A\subseteq B. A\subseteq B. e.g. (2,4] (2,4] is a superset of (2,4) (2,4) so (2,4]\supseteq (2,4) (2,4]\supseteq (2,4) and in fact (2,4]\supset(2,4) (2,4]\supset(2,4) since 4\notin (2,4) 4\notin (2,4).
    The union and intersection operators are commutative and associative. Surprisingly, each of these operations is distributive over the other:
    Commutative: A\cup B = B \cup A A\cup B = B \cup A and A \cap B = B \cap A A \cap B = B \cap A 
    Associative: A\cup (B \cup C) = (A \cup B) \cup C A\cup (B \cup C) = (A \cup B) \cup C and A\cap (B \cap C) = (A \cap B)\cap C A\cap (B \cap C) = (A \cap B)\cap C  
    Distributive: A \cup \left( B \cap C \right) = (A \cup B) \cap (A \cup C), \; A \cup \left( B \cap C \right) = (A \cup B) \cap (A \cup C), \; and    A \cap \left( B \cup C \right) = (A \cap B) \cup (A \cap C) A \cap \left( B \cup C \right) = (A \cap B) \cup (A \cap C)  
    De Morgan’s laws involve taking the complement of a union or intersection of sets:
    \bullet\bullet (A\cup B)^{C}=A^{C}\cap B^{C} (A\cup B)^{C}=A^{C}\cap B^{C}
    \bullet\bullet (A \cap B)^{C}=A^{C}\cup B^{C} (A \cap B)^{C}=A^{C}\cup B^{C}
    Set algebra involves these three operations: intersection, union and complement. The universe UU and the empty set \emptyset \emptyset have the following important properties when we take intersection or union with another set AA.
    \displaystyle A \cap U = A, \quad A \cup U = U , \quad A \cup \emptyset = A, \quad \displaystyle A \cap U = A, \quad A \cup U = U , \quad A \cup \emptyset = A, \quad and \displaystyle A \cap \emptyset = \emptyset \; . \displaystyle A \cap \emptyset = \emptyset \; .
    We can use Venn diagrams to provide visualization of set operations. We use a box to represent the universe U U, and within that box a circle or ellipse to represent a “ fence" enclosing a set: