Set Theory and Numbers

Set notation should be familiar to the reader. Recall x ∈ A means that x is an element of A; the negation is $$x\notin A$$ . Notation for every member of a set A belonging to a set B is $$A \subseteq B$$ or $$B \supseteq A$$ . We say that A is a subset of B or B is a superset superset of A. If A is a proper subset of B, that is, $$A \subseteq B$$ but $$A\not =B$$ , then one can write $$A \subset B$$ . To prove that two sets A and B are equal, it is necessary to show that each is contained in the other; that is, $$A \subseteq B$$ and $$B \subseteq A$$ . Familiar sets are the empty set $$\varnothing$$ , which contains no elements and is a subset of every set, the natural numbers $$\mathbb{N} =\{ 1,2,3,\cdots \,\}$$ , the integers $$\mathbb{Z} =\{ \cdots \,,-2,-1,0,1,2,\cdots \,\}$$ , the rational numbers $$\mathbb{Q}$$ , the real numbers $$\mathbb{R}$$ , and the complex numbers $$\mathbb{C}$$ .