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.

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  1. V

    Are both sample spaces the same or do they mean different things?

    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...
  2. V9999

    I May I use set theory to define the number of solutions of polynomials?

    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...
  3. S

    Specifying vertical asymptotes in periodic functions in set notation

    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...
  4. E

    Proof by Induction of shortest suffix of concatenated string

    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...
  5. bornofflame

    [LinAlg] Describe Ker(T) using set notation for T:P^1 -> R

    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}##...
  6. bornofflame

    [LinAlg] Set Notation, Subspaces, Bases, Ranges, & Kernels

    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...
  7. Z

    Is the Notation Implied for a Singleton Set or Can It Have Multiple Elements?

    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...
  8. R

    Set Notation Question: Converting to Set Builder Notation | Homework Help

    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...
  9. Derek Hart

    B Proof that (x^n)/n has a limit of 0 at infinity

    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...
  10. M

    I What does N^2 mean in the case of natural numbers?

    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?
  11. M

    MHB Shaded region in terms of set notation

    I'm having trouble in expressing the shaded region in set notation (Thinking) Many Thanks :)
  12. pellman

    B What does this set notation mean?

    $$ A \oplus B $$ where A and B are sets
  13. amilapsn

    Proving or Disproving a Statement in Set Notation

    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...
  14. S

    Translating set notation to english

    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...
  15. E

    Help with some unfamiliar set notation

    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...
  16. reenmachine

    Set Notation , Set Theory

    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...
  17. reenmachine

    Set Theory , building a set notation

    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...
  18. reenmachine

    Building a set notation part II

    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...
  19. reenmachine

    What is the set notation for a sequence with a common difference of 3/4?

    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...
  20. B

    Need help with writing set notation.

    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...
  21. E

    Use set notation to define the language generated by the grammar

    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'...
  22. E

    Understanding Set Notation: Exploring the Relationship Between Sets S and W

    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...
  23. S

    Basic set notation question.

    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...
  24. S

    Proving Power Set Notation: Let B \subseteq U

    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...
  25. G

    Set Notation Question Very Easy

    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...
  26. D

    Set notation (union and intersection)

    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...
  27. E

    Describing all unique 2-tuples from a set using Set Notation

    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...
  28. N

    Writing a r.v. in set notation

    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...
  29. M

    Set Notation in Probability: Clarifying the Meaning of Union and Intersection

    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...
  30. B

    Finding Range of Set: Notation & Shorthand

    *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...