Math Problem Statement

pourquoi pour une partei de p element on peut generer 2^p elemen

Solution

When you have a set with pp elements, the reason you can generate 2p2^p subsets comes from the concept of binary choices. Here's why:

Explanation:

For each element in the set, there are two possibilities: either the element is included in the subset, or it is not included. So for each element, you have 2 choices.

  • For the first element, you can either include it or exclude it (2 choices).
  • For the second element, you can also either include it or exclude it (2 choices).
  • This process continues for all pp elements.

The total number of different subsets is the product of the choices for each element. Since each element has 2 possible outcomes (included or not), the total number of subsets is:

2×2×2××2=2p2 \times 2 \times 2 \times \cdots \times 2 = 2^p

Thus, a set with pp elements has 2p2^p subsets, including the empty set (where no elements are chosen) and the set itself (where all elements are chosen).

Example:

For a set with 3 elements, say {a,b,c}\{a, b, c\}, the possible subsets are:

  • \emptyset (the empty set)
  • {a}\{a\}, {b}\{b\}, {c}\{c\}
  • {a,b}\{a, b\}, {a,c}\{a, c\}, {b,c}\{b, c\}
  • {a,b,c}\{a, b, c\}

In total, there are 23=82^3 = 8 subsets.

Key Idea:

The number 2p2^p arises because for each of the pp elements, we make an independent binary choice (include or exclude). These choices multiply, giving 2p2^p possible combinations, or subsets.

Would you like a more detailed explanation or an example with more elements? Here are five related questions you could explore:

  1. What is the significance of the empty set when counting subsets?
  2. How is the binomial theorem related to counting subsets?
  3. Can you explain the difference between a power set and a set of subsets?
  4. How do you compute the number of subsets with exactly kk elements?
  5. What is the relation between binary numbers and the number of subsets?

Tip: The set of all subsets of a given set is called the power set, and its size is always 2p2^p for a set of pp elements.

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Math Problem Analysis

Mathematical Concepts

Set Theory
Combinatorics
Power Sets

Formulas

2^p (number of subsets of a set with p elements)

Theorems

Power Set Theorem

Suitable Grade Level

Grades 9-12