Topic 1.2 Mathematical Sets Flashcards
What information should come to mind when faced with this set (N)?
That symbol denotes the set of natural numbers/non-standard integers e.g. {1, 2, 3, 4…}
What information should come to mind when faced with this set (Z)?
That symbol denotes the set of integers, which consist of the positive and negative whole numbers as well as zero e.g. {… -3, -2, -1, 0, 1, 2, 3…}
What information should come to mind when faced with this set (Q)?
That symbol denotes the set of rational numbers (recall, every integer is a rational number, as it can be expressed as a fraction with denominator = 1). This set consists of numbers that can be written as fractions m/n where m and n are integers e.g. 1/2, 78/23, (−3)/4, 0/5
What information should come to mind when faced with this set (R)?
That symbol denotes the set of real numbers (although, that could lead one to question what counts as a real number at all). The set consists of all the rational numbers together along with all the irrational numbers.
Provide a definition for real numbers
Any limits of sequences of real numbers are real numbers
How would the following statement read? x∈A
‘x’ is in set A
How would the following statement read? π∉Q
‘π’ is not in set Q
In what ways do sets vary from lists?
They vary in terms of order and repetition i.e. order/repetition doesn’t matter in sets; order/repetition matters in lists
How would you represent a tuple?
With rounded brackets e.g. (a, b, c), where (a, b, c) ≠ (a, c, b) ≠ (a, a, b, c)
Why is the following reasonable? {0, 0, 0, 1, 2, 7, 9} = {0, 1, 2, 7, 9}
Sets contain unique objects, which is why repetition and order don’t matter
Why are the following incorrect? 2⊆Z {2}∈R
Because these statements imply that there are subsets in sets, sets that have specified no such thing
If A and B are sets, when would A = B
When A⊆B & B⊆A
How would one read the following statement? S={x∈R|sin(x)>0}
The line in the middle is read as ‘such that’. According to the statement, the set should only contain elements of R that obeyed the condition sin(x)>0
What’s the difference between open and closed brackets?
Closed brackets [] means the element it encloses is put into consideration, whereas open brackets () means the element it encloses isn’t put into consideration e.g. a is included but b isn’t [a, b)
Let A be the set containing all integer multiples of π. Express A using set builder notation
A={x∈R|∃k∈Z x=kπ}
Let A be the set containing all integer multiples of π. Express A using abbreviated set builder notation
A={kπ|k∈Z}
What is the abbreviated set builder notation for the set of rational numbers?
Q={m/n|m,n∈Z, n≠0}
How do you express when all elements of set A are also elements of set B and vice versa?
A = B
How do you know when A is a subset of B?
When every element of A is an element of B
How do you express that A is a subset of B? What does this mean?
Expressed as A⊆B
This means that if x⊆A, then x⊆B (∀x⊆A x⊆B)
Is the following true? Q∈R
No. Rational numbers are a subset of real numbers, they are not ‘in’ real numbers i.e. Q⊆R
Assume A and B are two distinct sets. What does A∪B mean?
A∪B means the union of sets A and B, whereby the union is a set of elements that are in at least one of A or B (x∈A∪B if and only if x∈A or x∈B)
Assume A and B are two distinct sets. What does A∩B mean?
A∩B means the intersection of sets A and B, forming a set comprised of the elements that are in both A and B (x∈A∩B if and only if x∈A and x∈B aka A∩B={x∈A│x∈B}={x∈B│x∈A})
What is the notation for empty sets? Provide an example
{} = ∅
(−1,2)∩(100,101)=∅
Let A and B be two distinct sets. What is the Cartesian product of A and B and how would you denote it?
The Cartesian product is the set of all possble ordered pairs we can build using elements from a defined set, in this case, we would use elements of A as the first element in the ordered pair, and we would use elements of B as the second element in the pair.
A ×B={(a,b)|a∈A,b∈B}
