Section 1.1 Definitions
Definitions play a very important role in mathematics. A direct proof of a proposition in mathematics is often a demonstration that the proposition follows logically from certain definitions and previously proven propositions. A definition is an agreement that a particular word or phrase will stand for some object, property, or other concept that we expect to refer to often. In many elementary proofs, the answer to the question, “How do we prove a certain proposition?”, is often answered by means of a definition. For mathematical proofs, we need very precise and carefully worded definitions.
Subsection Definitions Involving the Integers and the Real Numbers
Definition.
An integer \(a\) is an even integer provided that there exists an integer \(n\) such that \(a = 2n\text{.}\) An integer a is an odd integer provided there exists an integer \(n\) such that \(a = 2n + 1\text{.}\)
Definition.
A nonzero integer \(m\) divides an integer \(n\) provided that there is an integer \(q\) such that \(n = m \cdot q\text{.}\) We also say that \(m\) is a divisor of \(n\text{,}\) \(m\) is a factor of \(n\text{,}\) and \(n\) is a multiple of \(m\text{.}\) The integer 0 is not a divisor of any integer. If \(a\) and \(b\) are integers and \(a \ne 0\text{,}\) we frequently use the notation \(a \mid b\) as a shorthand for “a divides b.”
Definition.
A natural number \(p\) is a prime number provided that it is greater than 1 and the only natural numbers that are factors of \(p\) are 1 and \(p\text{.}\) A natural number other than 1 that is not a prime number is a composite number. The number 1 is neither prime nor composite.
Definition.
Let \(n \in N\text{.}\) If \(a\) and \(b\) are integers, then we say that \(a\) is congruent to \(b\) modulo \(n\) provided that \(n\) divides \(a - b\text{.}\) A standard notation for this is \(\modulo{a}{b}{n}\text{.}\) This is read as “\(a\) is congruent to \(b\) modulo \(n\)” or “\(a\) is congruent to \(b\) mod \(n\)”.
Definition.
For \(x \in R\text{,}\) we define \(\abs{x}\text{,}\) called the absolute value of \(x\), by
Subsection Definitions Involving Sets
Definition.
A set \(T\) that is a subset of \(\Z\) is an inductive set provided that for each integer \(k\text{,}\) if \(k \in T\) , then \(k + 1 \in T\text{.}\)
Definition.
Two sets, \(A\) and \(B\text{,}\) are equal when they have precisely the same elements. In this case, we write \(A = B\) . When the sets \(A\) and \(B\) are not equal, we write \(A \ne B\text{.}\)
The set \(A\) is a subset of a set \(B\) provided that each element of \(A\) is an element of \(B\text{.}\) In this case, we write \(A \subseteq B\) and also say that \(A\) is contained in \(B\text{.}\) When \(A\) is not a subset of \(B\text{,}\) we write \(A \not \subseteq B\text{.}\)
Definition.
Let \(A\) and \(B\) be two sets contained in some universal set \(U\text{.}\) The set \(A\) is a proper subset of \(B\) provided that \(A \subseteq B\) and \(A \ne B\text{.}\) When \(A\) is a proper subset of \(B\text{,}\) we write \(A \subset B\text{.}\)
Definition.
Let \(A\) and \(B\) be subsets of some universal set \(U\text{.}\) The intersection of \(A\) and \(B\text{,}\) written \(A \cap B\) and read “\(A\) intersect \(B\text{,}\)” is the set of all elements that are in both \(A\) and \(B\text{.}\) That is,
The union of \(A\) and \(B\text{,}\) written \(A \cup B\) and read “\(A\) union \(B\text{,}\)” is the set of all elements that are in \(A\) or in \(B\text{.}\) That is,
Definition.
Let \(A\) and \(B\) be subsets of some universal set \(U\text{.}\) The set difference of \(A\) and \(B\text{,}\) or relative complement of \(B\) with respect to \(A\text{,}\) written \(A - B\) and read “\(A\) minus \(B\)” or “the complement of \(B\) with respect to \(A\text{,}\)” is the set of all elements in \(A\) that are not in \(B\text{.}\) That is,
The complement of the set \(A\text{,}\) written \(A^c\) and read “the complement of \(A\text{,}\)” is the set of all elements of \(U\) that are not in \(A\text{.}\) That is,