What is Subsets: Definition and 220 Discussions

In mathematics, a set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment). A is a subset of B may also be expressed as B includes (or contains) A or A is included (or contained) in B.
The subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the join and meet are given by intersection and union, and the subset relation itself is the Boolean inclusion relation.

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  1. PhysicsRock

    I Proof about pre-images of functions

    The problem reads: ##f:M \rightarrow N##, and ##L \subseteq M## and ##P \subseteq N##. Then prove that ##L \subseteq f^{-1}(f(L))## and ##f(f^{-1}(P)) \subseteq P##. My co-students and I can't find a way to prove this. I hope, someone here will be able to help us out. It would be very...
  2. PhysicsRock

    I Is the solution to this problem as trivial as I think?

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  3. Andrew1235

    Finding all subsets of a list of positive integers using backtracking

    The following Python 3 code is provided as the solution to this problem (https://leetcode.com/problems/subsets/solution/) that asks to find all subsets of a list of integers. For example, for the list below the output is [[], [1], [1, 2], [1, 2, 3], [1, 3], [2], [2, 3], [3]]. I am not familiar...
  4. MidgetDwarf

    Is it possible to make graphs of subsets of Rational Numbers in Mathem

    Is it possible to make subsets of rational numbers in Mathematica using the plot command, or any other command? Ie., say I want to graph the set of rational numbers from 0 to 1.
  5. M

    MHB How is lexicographic order used in ranking and unranking subsets?

    Hey! :giggle: I am looking at the following codes: It is lexicographic order related to ranking and unranking. Here is also an example: There is also the Gray code: with the repective examples: I haven't really understood the ranking and the unranking. So we have a set and...
  6. A

    How many subsets are in {∅} and {0}?

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  7. M

    MHB Subsets of permutation group: Properties

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  8. Math Amateur

    I Closed Subsets in a Toplogical space ....

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  9. Math Amateur

    MHB Countably Dense Subsets in a Metric Space .... Stromberg, Lemma 3.44 .... ....

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  10. Math Amateur

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  11. Math Amateur

    MHB Open Subsets in a Metric Space .... Stromberg, Theorem 3.6 ... ....

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  12. Math Amateur

    MHB Compact Subsets of R .... Sohrab, Proposition 4.1.8 .... ....

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  13. Math Amateur

    I Compact Subsets of R .... Sohrab, Proposition 4.1.8 .... ....

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  14. Math Amateur

    I Compact Subsets of R .... Sohrab, Proposition 4.1.1 (Lindelof) ....

    I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 4: Topology of ##\mathbb{R}## and Continuity ... ... I need help in order to fully understand the proof of Proposition 4.1.1...Proposition 4.1.1, some preliminary notes and its proof read...
  15. Math Amateur

    MHB Compact Subsets of R .... Sohrab, Proposition 4.1.1 (Lindelof)

    I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 4: Topology of \mathbb{R} and Continuity ... ... I need help in order to fully understand the proof of Proposition 4.1.1...Proposition 4.1.1, some preliminary notes and its proof read as...
  16. Eclair_de_XII

    How many subsets of size three or less are in a n-object set

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  17. S

    MHB Proper Subsets and Relations of Sets

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  18. M

    MHB Exploring Basis Subsets in a 5-Element Vector Space

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  19. Math Amateur

    MHB Understanding Proper Subsets of Ordinals in Searcoid's Theorem 1.4.4 - Peter

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  20. Math Amateur

    I Proper Subsets of Ordinals .... .... Another Question .... ....

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  21. Math Amateur

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  22. Math Amateur

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  23. ubergewehr273

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  24. U

    Proving S is a Subset of T in R³

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  25. Math Amateur

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  26. lfdahl

    MHB Find the maximal number of subsets, k.

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  27. SSequence

    B Subsets of Rational Numbers and Well-Ordered Sets

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  28. F

    I Sets, Subsets, Possible Relations

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  29. sa1988

    Are the following subsets open in the standard topology?

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  30. M

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  31. NoName3

    MHB Proving Subsets of Intervals in $\mathbb{R}$

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  32. G

    MHB Subspace spanned by subsets of polynomials

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  33. C

    Clarification: proof that perfect subsets of R^k uncountable

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  34. W

    Inner Joins between Subsets of the Same Table.

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  35. O

    MHB Exploring Limits and Infinite Subsets of $\Bbb{N}$

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  36. O

    MHB Common Fixed Point Theorems on Weakly Compatible Maps

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  37. Math Amateur

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  38. R

    Prove of number of subsets of a set

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  39. Shackleford

    Show converges uniformly on compact subsets of C

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  40. evinda

    MHB The subsets of finite sets are finite sets.

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  41. C

    Proof Linear Span Subsets: Proving L(S) is Smallest Subspace of V

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  42. H

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  43. D

    How to Use Counting Methods and Inclusion-Exclusion for Subsets?

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  44. L

    How many subsets are there of a set consisting of n elements?

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  45. M

    Topology generated by a collection of subsets of ##X##

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  46. M

    Describing closures and interiors of subsets of Moore Plane

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  47. B

    Infimum of Subsets in R: True or False?

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  48. C

    MHB Cardinality, Subsets, Sequences

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  49. V

    Min/max of functions restricted to subsets of their domains

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  50. evinda

    MHB With how many ways can we choose disjoint subsets?

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