Chapter 1 Notation & terminology Flashcards
1.1 Sets
What is a set?
A set is a collection of objects.
1.1 Sets
What is an element?
element of a set?
Each object in a set is said to be an element of the set.
1.1 Sets
What do round brackets and square brackets indicate in a set?
for example [0,10)
A round bracket indicases that the endpoint is not included in the set.
A square bracket indicates that the endpoint is included in the set.
[0,10) is the set of values that 0 0,10) is the set of values of x such
1.1 Sets
What does ∅ mean?
The empty set, ie. the set that contains no elements
1.1 Sets
What does ∈ mean?
Is a member of the set
if S is the set of even numbers, then 2∈S
1.1 Sets
What does ∉ mean?
Is not a member of the set
if S is the set of even numbers, then 13∉S
1.1 Sets
What does ⊂ mean?
Is a subset of
C⊂D if every element of the set C is also an element of the set D
1.1 Sets
What does ⊄ mean?
Is not a subset of
C⊄D means that the set C contains at least one element that is not an el
1.1 Sets
What does ∪ mean?
Union (Or)
C∪D is the set of elements contained in set C or set D or both
1.1 Sets
What does ∩ mean?
Intersection (And)
C ∩ D is the set of elements in set C that are also elements of set D
1.1 Sets
What does Ā mean?
Complement of set A
Ā is the set of elements that do not belong to set A
1.1 Sets
What does A’ mean?
Alternative notation for the complement of set A
complement being the set of elements that do not belong to set A
1.1 Sets
What does ℕ mean?
The set of natural numbers
ℕ = {1,2,3,4,…]
1.1 Sets
Should 0 be included in ℕ?
We use the convention that 0 is not a member of this set
Some mathematicians do define the set of natural number to include 0
1.1 Sets
What does ℤ mean?
The set of intergers
ℤ = {…,-3,-2,-1,0,1,2,3,…}
1.1 Sets
What does ℚ mean?
The set of rational number (fractions)
the set numbers of the form p/q, where p,q ∈ ℤ and q ≠ 0
1.1 Sets
What does ℝ mean?
The set of real numbers
the set of all number between - ∞ and ∞
1.1 Sets
What does the superscripts + or - mean in situations such as ℤ+ or ℤ-
In the example the + and - should be superscript
the + or - refers to the positive or negative numbers
with in the set respectivally
ℤ = {…,-3,-2,-1,0} = ℕ
1.2 Logic & Proofs
What does ∀ mean?
For all
(Xsquared) 1> ∀>x 1
1.2 Logic & Proofs
What does ∃ mean?
There exists
∃ x ∈ ℝ such that x+1= 5
1.2 Logic & Proofs
What does ∄ mean?
There does not exist
∄ x ∈ ℕ such that xsquared = 2
1.2 Logic & Proofs
What does : mean?
Such that
ℝ−= {x : - ∞< X<0}
1.2 Logic & Proofs
What does st mean?
Such that
∃x∈ ℝ st x + 1 = 5
1.2 Logic & Proofs
What does ⇒ mean?
Implies
x = -2 ⇒ xsquared = 4