• Real Analysis
• 1. Sets and Relations
• 2. Infinity and Induction
• 2.1. Countable Infinity
• 2.2. Uncountable Infinity
• 2.3. The Principle of Induction
• 2.4. The Real Number System
• 2.5. Pi and e are irrational
• 3. Sequences of Numbers
• 4. Series of Numbers
• 5. Topology
• 6. Limits, Continuity, and Differentiation
• 7. The Integral
• 8. Sequences of Functions
• 9. Historical Tidbits
• Java Tools

Examples 2.2.4(b):

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To show that the cardinality of P(S) is greater than or equal to that of a set S we have to find either a surjection from P(S) to S or an injection from S to P(S).

Define the function f from S to P(S) as follows:

• f(s) = {s} for any s in S

i.e. every element s in S is mapped to the set {s} containing the single element s. Note that the set {s} is an element of the power set P(S).

This map is clearly one-to-one, and therefore card (S) card(P(S)).

This does not prove that card(P(S)) > card(S). However, that statement is also true, but its proof is more complicated.

Interactive Real Analysis, ver. 2.0.1(b)
(c) 1994-2017, Bert G. Wachsmuth
Page last modified: Mar 3, 2017