Definitions Flashcards
definition
An exact statement of the meaning of a word
theorem
A proved mathematical statement
lemma
A proved result, typically less important than a theorem; often used to prove theorems.
proposition
A proved statement, typically less important than a lemma
corollary
The direct result of a theorem, lemma, or proposition
axiom
A statement assumed to be true.
set
A well-defined collection of objects
ℕ
{0, 1, 2, 3, …}
ℤ
{…, -2, -1, 0, 1, 2, …}
ℤ+
{ 1, 2, 3, 4, 5, …}
ℚ
{ a/b | a,b∈ℤ, b ≠ 0}
ℚ’ / ℍ
{ x | x is a decimal number that neither terminates or repeats}
ℝ
{ x | x ∈ ℚ⋃ℚ’ }
ℂ
{ a + bi | a,b ∈ ℝ, i^2 = -1 }
The empty set
The set that contains no elements
A⋃B
{ x | x∈A, x∈B, or both }
A⋂B
{ x | x∈A and x∈B }
Ᾱ / A^c
{ x | x∉A }
A - B
{ x∈A | x∉B }
A⊆B
∀x, if x∈A, x∈B.
A⊂B
∀x, if x∈A, x∈B,
AND ∃ y∈B ∋ y∉A.
power set
Let A be a set.
Then the power set of A is the set containing all subsets of A; denoted P(A).
AxB
Let A,B be sets.
Then AxB = { (a,b) | a∈A, b∈B }
DeMorgan’s Laws
~(P and Q) = ~P or ~Q
~(P or Q) = ~P and ~Q
partition
Let X be a set.
A partition S of X is a collection of sets such that:
- If A,B∈S, then A⋂B = ∅, unless A=B, and
- ⋃(A∈S) A = X
statement
A sentence that is either true or false, and can be evaluated.
open statement
A statement with a variable involved
WFF
A well-formed formula.
A statement consisting of at least one statement and one connective.
equivalent
Two WFF are equivalent if they have the same truth values.
tautology
A statement that is always true.
contradiction
A statement that is always false.
∀
The universal quantifier
∃
The existential quantifier
Odd integer
A number n is odd if ∃ m∈ℤ ∋ n = 2m+1
Even integer
A number n is even if ∃ m∈ℤ ∋ n = 2m
Divides
Let a,b∈ℤ.
Then a | b if ∃ k∈ℤ ∋ ak = b
|a|
= a if a ≥ 0
-a otherwise
a ≡ b mod c
Let a,b,c ∈ℤ.
Then a ≡ b mod c , if c | (b-a)
e.g.
33 ≡ 3 mod 10 –> 103 R3
33 ≡ 13 mod 10 –> 102 R13
33 ≡ -7 mod 10 –> 10*4 R-7
a ≤ b
a ≤ b if ∃ c ≥ 0 ∋ a+c=b
a
a 0 ∋ a+c=b
prime number
A number p is prime if it has exactly 2 positive divisors
p∈ℤ+
composite number
Any number n∈ℤ+ (other than 1) which is not prime.
f’(x)
f’(x) = lim h→0 [ f(x+h) - f(x) / h ]
…as long as the limit exists!
local max
A function f has a local max at c if f(c) ≥ f(x) for ∀x close to c.
EVT (extreme value theorem)
Let f be continuous on [a,b]. Then f has an absolute max at some c∈[a,b].
Fermat theorem
Let f be continuous function and have a local max at c. If f’(c) exists, then f’(c) = 0.
Rolle’s Theorem
Let f be a continuous function satisfying
- f is cts on [a,b]
- f is differentiable on (a,b)
- f(a) = f(b).
Then ∃ c∈(a,b) ∋ f’(c) = 0.
(basically, if a function has two equal values at two distinct points, then there must be a stationary point between them, i.e. the slope of the tangent is 0)
Triangle inequality
a + b | ≤ | a | + | b |
If A is a set, what is |A|?
It is the number of elements in A, either finite or infinite, called the cardinality of A
If P(A) is the power set of A, what is the cardinality of P(A)?
|P(A)| = 2^|A|
What are the five basic connectives?
Let S and T be statement variables
- Negation of S (~S)
- Conjunction (S∧T)
- Disjunction (S∨T)
- Implication (S => T)
- Biconditional (S T)
What does biconditional mean?
Let S and T be statement variables
S T
S iff T
They imply each other, that is
S => T and T => S
What is a mathematical proof?
A formal series of statements showing that if a hypothesis is true, then necessarily ‘this’ conclusion must also be true
