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Chapter 2: Problem 46

Find the power set of the "tripleton" set \(\mathrm{S}=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}\\}\), and the number of elements of \(\mathrm{P}(\mathrm{S})\).

Short Answer

Expert verified
The power set of the set S = {a, b, c} is P(S) = { \(\emptyset\), {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} } and the number of elements in P(S) is 8.

Step by step solution

01

Identify all possible subsets of S

First, find all possible subsets of the set S = {a, b, c}. These are as follows: 1. Empty set: {}(also written as \(\emptyset\)) 2. Single element subsets: {a}, {b}, {c} 3. Two-element subsets: {a, b}, {a, c}, {b, c} 4. Three-element subset: {a, b, c} (which is equal to the set S)
02

Write out the power set of S

The power set is the set of all possible subsets. Based on the subsets we found in Step 1, we can write the power set as: P(S) = { {}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} }
03

Find the number of elements in P(S)

To find the number of elements in the power set P(S), simply count the number of subsets inside the set: There are 8 subsets within the power set, hence, the number of elements in P(S) is 8.

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Most popular questions from this chapter

Find \(A-B\) and \(A-(A \cap B)\) for \(\mathrm{A}=\\{1,2,3,4\\} \quad\) and \(\quad \mathrm{B}=\\{2,4,6,8,10\\}\)

(1) Find set \(\mathrm{S}=\mathrm{A} \cup(\mathrm{B} \cap \mathrm{C})\) where \(\mathrm{U}=\\{2,4,6,8,10, \mathrm{x}, \mathrm{y}, \mathrm{z}\\}, \quad \mathrm{A}=\\{2,4, \mathrm{x}, \mathrm{y}\\}\) \(\mathrm{B}=\\{2,4,6,8,10\\}, \quad\) and \(\quad \mathrm{C}=\\{6,8, \mathrm{z}\\}\) (2) Draw the Venn Diagram of the set \(A \cup(B \cap C)\).

List all the subsets of \(\mathrm{C}=(1,2)\).

Prove that the power set \(\mathrm{P}(\mathrm{A})\) of any set \(\mathrm{A}\) of \(\mathrm{n}\) elements contains exactly \(2^{\mathrm{n}}\) elements.

Illustrate one of De Morgan's Theorems with the use of Venn Diagrams.
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