Basic Set Theory: fun and games with the power set

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Daniel Mansfield

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Name	Status	Author	Last Modified
Basic Set Theory: fun and games with the power set	Ready to use	Daniel Mansfield	02/06/2019 06:57
Power set - empty set and \|P(P(P(S))\|	draft	Lois Rollings	13/08/2020 09:32
Simon's copy of Basic Set Theory: fun and games with the power set	draft	Simon Thomas	07/03/2019 14:23

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Question statement

We use the notation $\left|A\right| = n$ to mean that $A$ contains $n$ elements. This is often called the cardinality of the set. Here is a handy fact about the number of elements in a power set.

If $\left|A\right| = n$ then $\left|P(A)\right| =2^n$ .

Using this fact, answer the following questions regarding the power set.

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Since $P(\left\{\right\}) = 2^{\left|\left\{\right\}\right|} = 2^0 = 1$ we know that $P(\left\{\right\})$ is a one-element set which contains $\left\{\right\}$ . There is only one possible answer

$\left\{\left\{\right\}\right\}$ .

We construct the answer gradually. Since $\left|S\right| = \var{n}$

$\left|P(S)\right| = 2^\var{n}$
$\left|P(P(S))\right| = 2^{\left|P(S)\right|} = 2^{2^\var{n}}$
$\left|P(P(P(S)))\right| = 2^{\left|P(P(S))\right|} = 2^{2^{2^\var{n}}}$ .

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This question is used in the following exams:

Possible questions by Lois Rollings in Algebra 1 - Possibilities.
Joshua's copy of Basic Set Theory by Joshua Capel in Joshua's workspace.
Week 1: Basic Set Theory by Daniel Mansfield in Discrete Mathematics.
Week 1: Basic Set Theory - V2 by Sean Gardiner in Discrete Mathematics.
Shivendra's copy of Week 1: Basic Set Theory - V2 by Deactivated user in Deactivated user's workspace.
Shaheen's copy of Week 1: Basic Set Theory - V2 by Shaheen Charlwood in Shaheen's workspace.

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Basic Set Theory: fun and games with the power set

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Daniel Mansfield 7 years, 9 months ago

Daniel Mansfield 7 years, 9 months ago

Daniel Mansfield 7 years, 10 months ago

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Generated value: `integer`

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Basic Set Theory: fun and games with the power set

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Daniel Mansfield 7 years, 9 months ago

Daniel Mansfield 7 years, 9 months ago

Daniel Mansfield 7 years, 10 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Mathematical expression

Advanced settings

Checking accuracy

Variable value generators

Pattern restriction

Variables

String restrictions

Generated value: `integer`