What is Set notation: Definition and 31 Discussions
In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by enumerating its elements, or stating the properties that its members must satisfy.Defining sets by properties is also known as set comprehension, set abstraction or as defining a set's intension.
I came up with two different forms of the sample space S, but I am not sure if they mean the same thing or the first one could mean something different. H stands for heads showing up and T stands for tails showing up.
$$ S = \{ \{i,j\}: i \in \{H,T\}, j \in \{H,T\} \} $$
$$ S = \{ (i,j) : i...
Let ##Q_{n}(x)## be the inverse of an nth-degree polynomial. Precisely,
$$Q_{n}(x)=\displaystyle\frac{1}{P_{n}(x)}$$,
It is of my interest to use the set notation to formally define a number, ##J_{n}## that provides the maximum number of solutions of ##Q_{n}(x)^{-1}=0##. Despite not knowing...
Hi all,
What is the general set notation for specifying a vertical asymptote and domain for a periodic function? For example, if I have a periodic function which has a period of pi/2, and within that period, a vertical asymptote occurs at pi/4. The domain is R, excluding that vertical...
Homework Statement
Wherein α, β are strings, λ = ∅ = empty string, βr is the shortest suffix of the string β, βl is the longest prefix of the string β, and T* is the set of all strings in the Alphabet T, |α| denotes the length of a string α, and the operator ⋅ (dot) denotes concatenation of...
Homework Statement
2. Let ##T: \mathbb{P}^1 \rightarrow \mathbb{R} \text{ be given by } T(p(x)) = \int_0^1 p(x)~dx## Describe ##Ker(T)## using set notation.
Homework Equations
##p(x) \in \mathbb{P}^1~|~ p(x) = a_0 + a_1x##
The Attempt at a Solution
##T: \mathbb{P}^1 \rightarrow \mathbb{R}##...
Homework Statement
I don't want to clog up the forums with a few "small" problems so I am lumping them together here.
2. Let ##T:P^1 → R \text { be given by } T(p(x)) = \int^b_a p(x)dx##. Describe Ker(T) using set notation.
3. Let ##H = \left\{f ∈ C[a, b] | f'(x) ≥ 0 ~\text for...
Homework Statement
For the following notation,
D \subseteq \{ z | \exists x,y \in S (z = x - y ) \}
I'm wondering if S can only have one element, or does the notation imply that | S | > 1
Homework EquationsThe Attempt at a Solution
For example, if I have
S = \{ 3 \}
is z = 3 - 3?
Or...
Homework Statement
So I have the set
{...(1/8),(1/4),(1/2),(1),(2),(4),(8)...}
I am suppose to put it in set builder notation..
Homework EquationsThe Attempt at a Solution
my answer was { x = {\frac {1}{2^n} : n \in ℤ } }
but my books was
{ x = {{2^n} : n \in ℤ } }
I understand both...
I understand that the standard proof is a bit different from my own, but I want to know if my reasoning is valid. PROOF:
Firstly, I assume that x is positive.
I then consider p = inf{n∈ℕ : n>x} . In other words, I choose "p" to be the smallest natural number greater than x. If we choose n>p...
What does the N^2 mean in this case? (Image below)
Does it mean, for all two pairs of natural numbers, a and b?
How would I represent non pair numbers, i.e. how would I write "For integers k,l, and m such that k>1, l>2, m>k+l" all in one line?
Homework Statement
Prove or disprove the following
(i) ##\forall a\in\mathbb{R}[(\forall \epsilon>0,a<\epsilon)\Leftrightarrow a\leq 0]##
2. The attempt at a solution
Can't we disprove the above statement by telling ##a\leq 0 \nRightarrow (\forall \epsilon>0,a<\epsilon)## through a counter...
I've been trying to think of the grammatically correct way to translate A\cupB and A\capB.
So, let's say A is the set of all animals and B is the set of all boats.
Then, A\cupB is the set of all entities which are either animals or boats (or both).
And A\capB is the set of entities...
I am reading http://walrandpc.eecs.berkeley.edu/126notes.pdf on the theory of random processes. The authors are making use of some unfamiliar notation early on, and I don't want to move on without understanding their formalisms.
First one is the := operator and a union operator that looks like...
Homework Statement
Build a notation for the set: ##\{ ... , -8 , -3 , 2 , 7 , 12 , 17 ,...\}##
Homework Equations
##2+5(0) = 2##
##2+5(1) = 7##
##2+5(-1) = -3##
etc...
The Attempt at a Solution
##\{ \ 2+5y \ | \ y \in \mathbb{Z} \ \}##
Take note that you could replace ##2## by any...
Homework Statement
Build a set notation for $$\bigcup_{i \in N}R × [i , i + 1]$$
The Attempt at a Solution
##\{(x,y) \in R : x \in R \ \ \exists z \in N \ \ z ≤ y ≤ (z+1)\}##
Last time I tried one of these kind of sets I struggled quite a bit , so I'm interested in knowing how much...
Homework Statement
As an exercise , the book I'm reading ask me to build a set notation for the following set:
##\{... \ , \frac{1}{27} \ , \frac{1}{9} \ , \frac{1}{3} \ , 1 \ , 3 \ , 9 \ , 27 , \ ...\}##
The Attempt at a Solution
After playing with the numbers a couple of minutes , I came...
Homework Statement
The book I'm reading ask me to build a set notation for the following set: ##\{-3/2 , -3/4 , 0 , 3/4 , 3/2 , 9/4 , 3 , 15/4 \}##.
The Attempt at a Solution
I attempted to build a set notation and came with the result:
##\{ x \in R : \exists y\in Z \ \ y(3/4)= x\}##
My...
I need help with some set notation. I am an engineering major and don't know the proper notation. I am trying to write two statements. I have two sets, x_1 = {1,2,3}, and x_2 = {4,5,6}.
Verbal Statements:
y_1 was measured for all values of x_1 and x_2
y_2 was measured, for all values of...
Homework Statement
As the topic states, I need to define a language, for a grammar, using set notation. Homework Equations
Here is the grammar:
S -> aaSB | λ
B -> bB | b
The Attempt at a Solution
Okay, I know that this creates strings that are either empty or consist of an even number of 'a'...
Ok, I'm not very familiar with set notation. I was just wondering if the following is correct notation and means what I think:
{x\inS : \existsy\inx, y\inW}
S is a set of pairs of symbols (tuples of length 2 is the technical term I believe). W is a set of symbols.
What I want is the...
G'day all,
If i let S={[aij]nxn, where aij = aji, n\inZ+}
How do I distinguish between a single natural number n or all natural numbers?
I am trying to show that for each n, this set makes a subspace, but am not sure how to represent this set without the ambiguity, as this is not a subspace...
my actual problem is to let B be a subset of the set U and prove
P(B^{C}_{U}) \neq (P(B))^{C}_{P(U)}
but I am confused on the scripts and not quite sure what they are wanting me to do
i have Let B \subseteq U where B = {b} and U = {B}
I know P(B) = {empty set, {b}} and P(U) = {empty...
Homework Statement
If the domain of a function was defined for all reals except undefined at say x=3 were there was a vertical asymptote would you right
(-infinity, 3)U(3, infinity)
I'm not sure if it is union or infinity it's union right because technically they don't intersect right...
Homework Statement
Find simpler notation for the two sets:
A= \bigcup^{\infty}_{j=0}[j,j+1]
and
B= \bigcap_{j \in Z}(R minus\ (j,j+1))
Homework Equations
The Attempt at a Solution
Not really sure what it means by "simpler notation"...
Does A=R since the union of...
Describing certain 2-tuples from a set using Set Notation
How do you describe, using set notation, a certain set of n-tuples generated from a set satisfying the following conditions
Condition 1: non of the components of an individual n-tuple can equal each other
Condition 2: if (a,b) is...
The definition I have for a random variable is
X=\lbrace \omega \in \Omega \vert X(\omega) \in B \rbrace \in F where F is a sigma algebra and B is a Borel subset of R.
Using function composition, how would one write a similar set notation definition for f(X), where f is a Borel measurable...
I'm having a little difficulty understaning the notation used in my probability class. Would someone tell me if I'm on the right track with it :)
A\cup B is the event "either A or B or both." Does this mean that A and B can happen together and that A or B can happen alone?
A\cap B...
*Suppose I want to find the range of the set \left\{ {t_1 ,t_2 , \ldots ,t_n } \right\} , that is, the difference between the maximum and minimum values (of the elements that is!) in the set.
Do I have to fully write out,
\max \left\{ {t_1 ,t_2 , \ldots ,t_n } \right\} - \min \left\{ {t_1...