Chapter 1, 2, and 3 MAT 305 Definitions and Axioms

Axiom 3.21 (The Axiom of Mathematical Induction)

Let P(n) be a statement about the natural number n. If: (i) P(1) is true, and (ii) P(k) implies P(k+1), then P(n) is true for every natural number n.

Definition of Prime (3.32)

A natural number p is prime iff p>1 and p has exactly two positive factors: 1 and p.

Definition of Fruvous (2.50)

A real number x is fruvous iff it is either the root of an E-F-O polynomial or can be written as x=log(subscript z)y where y is rational and z is an Osmodiar prime.

Definition of Statement (1.2)

A statement is a declarative sentence which is either true or false, but not both.

Definition 3.3 (Triomino)

A triomino is a truncated 2x2 board.

Definition of Separable (2.21)

An integer is separable iff (n-2k)(k-2n)=0 for some integer k.

Definition of Even (2.2)

An integer n is even iff n=2k for some integer k.

Definition of Odd (2.3)

An integer n is odd iff n=2k+1 for some integer k.

Definition of Uniquely Separable (2.24)

An integer n is uniquely separable iff there exists a unique integer k such that (n-2k)(k-2n)=0.

Definition for Division (2.27)

Let m and n be integers. We say that m divides n and write m|n iff there is an integer k such that n=k*m. In the case that m|n we can also say that n is divisible by m, or that m is a factor of n or that n is a multiple of m.

Axiom 2.25 (The Zero Product Property)

Let x and y be integers. Then xy=0 if and only if x=0 or y=0.

Axiom 2.12

Every integer is either even or odd.

Definition of Consecutive Integers (2.5)

For any integer n, we call the pair of integers n and n+1 consecutive.

Definition of Board (3.1)

For any positive integers m and n, the mxn board is a rectangle that has been subdivided into m rows and n columns of squares. An mxn board will contain mn squares. A truncated mxn board is an mxn board with a single corner square removed.

Axiom 1.11

If P and Q are statements then so are the following: 1. Not P 2. P and Q 3. P or Q 4. If P, then Q

Axiom 1.12

If P is a statement with a given truth value, then the negation of P, denoted -P, is a statement with the opposite truth value. We would read -P as "not P" or "the negation of P."

Axiom 1.15

Let P And Q be statements. Then the statement P or Q is true unless both P is false and Q is false.

Definition 1.36

Let P and Q be arbitrary statements. Consider the implication P->Q regardless of whether it is true or false. The converse of the implication P->Q is Q->P. The inverse of the implication P->Q is (-P)->(-Q). The contrapositive of the implication P->Q is (-Q)->(-P).

Axiom 1.27

Let P and Q be statements. Then the statement "If P, then Q" is true unless P is true and Q is false.

Axiom 1.14

Let P and Q be statements. Then the statement P^Q is true exactly when both P is true and Q is true, and is false otherwise.

Axiom 3.42 (Axiom of Strong Induction)

Let P(n) be a statement about the natural number n. If: (i) P(1) is true, and (ii) For each natural number k it is true that: If P(1) and P(2) and...P(k) are true, then P(k+1) is true. then P(n) is true for every natural number n.

Axiom 2.30

The only factors of the integer 1 are 1 and -1.

Definition of Parity (2.4)

The parity of an integer refers to its oddness or evenness. Two integers have the same parity iff they are both even or they are both odd.

Definition 3.5 (tiling)

We say that we can tile (or cover) a board with triominoes if it is possible to arrange some number of triominoes onto the board so that every square on the board is covered where no triominoes overlap and no parts of any triominoes hang over the edge of the board. In this case we also say that the board is tileable or coverable.