1 - Integers Flashcards
What is commutativity of addition (Axiom 1.1 (i))?
If m, n, and p are integers, then
m+n = n+m
What is associativity of addition (Axiom 1.1 (ii))?
If m, n, and p are integers, then
m+n)+p = m+(n+p
What is distributivity (Axiom 1.1 (iii))?
If m, n, and p are integers, then
m(n+p) = mn+mp
What is commutativity of multiplication (Axiom 1.1 (iv))?
If m, n, and p are integers, then
mn = nm
What is associativity of multiplication (Axiom 1.1 (v))?
If m, n, and p are integers, then
mn)p = m(np
Describe the identity element for addition (Axiom 1.2).
There exists an integer 0 such that whenever m E Z, m+0 = m
Describe the identity element for multiplication (Axiom 1.3).
There exists an integer 1 such that 1 ≠ 0 and whenever m E Z, 1(m) = m
Describe additive inverse (Axiom 1.4).
For each m E Z, there exists an integer, denoted by -m, such that m+(-m) = 0
Describe cancellation (Axiom 1.5).
Let m, n, and p be integers. If mn = mp and m ≠ 0, then n = p
What does the symbol E mean?
The symbol E means “is an element of”.
What does the symbol = mean?
Equals: the same number as.
What are 3 properties of the symbol =?
- Reflexivity
- Symmetry
- Transitivity
- Replacement
Describe reflexivity.
m = m
Describe symmetry.
If m = n, then n = m.
Describe transitivity.
If m = n and n = p then m = p.
Describe replacement.
If m = n, then m+p = n+p
What does ≠ mean?
Not equal to: m≠n means m and n are different.
Which properties of equalities does “≠” satisfy and what does it not satisfy?
Satisfies symmetry.
Doesn’t satisfy transitivity or reflexivity.
What does ∉ mean?
Not an element of.
What are axioms in mathematical language?
Truths or facts.
When m,nEZ, when do we say that m is divisible by n (or alternatively, n divides m)?
If there exists jEZ such that m = jn. We use the notation n|m (n divides m)
Define subtraction on Z.
m-n is defined to be m+(-n)
What do we assume throughout chapter 1 and 2?
We assume there is a set, denoted by Z, whose members we call integers. This set is equipped with binary operations called addition and multiplication.
What is a binary operation?
A binary operation on a set S is a procedure that takes two elements of S as input an gives another element of S as output.
