Question

Enumerate all the subsets of the set \(\\{5,6,7\\}\)

Short answer

The subsets are \(\{\}, \{5\}, \{6\}, \{7\}, \{5,6\}, \{5,7\}, \{6,7\}, \{5,6,7\}\).

Step-by-step solution

Understand the Concept of Subsets

The subsets of a set are combinations of its elements, including the empty set and the set itself. For a set with \(n\) elements, there are \(2^n\) possible subsets.

Calculate Total Number of Subsets

The given set \(\{5, 6, 7\}\) has 3 elements. Therefore, the total number of subsets is \(2^3 = 8\).

Enumerate All Subsets

Begin listing all possible combinations of elements from the set \(\{5, 6, 7\}\): 1. The empty set: \(\{\}\) 2. Single elements: \(\{5\}\), \(\{6\}\), \(\{7\}\) 3. Pairs: \(\{5, 6\}\), \(\{5, 7\}\), \(\{6, 7\}\) 4. The entire set: \(\{5, 6, 7\}\)

Key concepts

Combinatorics

In mathematics, combinatorics is the fascinating study of counting and arranging objects. It helps us understand how to count without actually having to list every possibility. One key concept within combinatorics is understanding subsets.
Subsets are essentially different groups or combinations you can form from a given set. For any set with size \(n\), the number of different subsets that can be formed is \(2^n\). This is because every element in the set can either be present or absent in a subset, leading to the exponential count.

• Example: For the set \(\{5, 6, 7\}\), which has three elements, it has \(2^3 = 8\) subsets.
• There are single-element subsets, two-element subsets, the full set itself, and even an empty set.

Combinatorics provides powerful methods like permutation and combination formulas to count arrangements in more complex scenarios. It forms the backbone for many areas in discrete mathematics, including probability and graph theory.

Set Theory

Set theory is a mathematical discipline focused on the study of sets, which are collections of distinct objects. Whether these objects are numbers, letters, or other entities, set theory provides a strict methodology to handle and manipulate collections.
In the context of subsets, set theory teaches us the basic properties of sets, including how to list and count different kinds of subsets. It emphasizes concepts such as:

• Null or Empty Set: A set with no elements, denoted by \(\{\}\).
• Universal Set: The set that contains all objects under consideration.
• Power Set: The collection of all subsets of a set \(S\), including the empty set and \(S\) itself.

For the set \(\{5, 6, 7\}\), set theory provides a framework to understand the complete listing of subsets mentioned in the solution. By using set notation and properties, students can systematically explore the possibilities contained within any set.

Mathematical Enumeration

Mathematical enumeration is a technique used to systematically count the number of ways certain arrangements or selections can be made. It involves methodically listing all possibilities to ensure nothing is missed.
When it comes to subsets of a set, enumeration helps us explicitly see the different combinations. With a set like \(\{5, 6, 7\}\), enumeration means listing each subset one by one:

• The first subset is the empty set: \(\{\}\).
• Then we have subsets containing single elements: \(\{5\}\), \(\{6\}\), and \(\{7\}\).
• Next, subsets with pairs of elements: \(\{5, 6\}\), \(\{5, 7\}\), \(\{6, 7\}\).
• Finally, the subset with all elements: \(\{5, 6, 7\}\).

This ensures we've accounted for all 8 potential subsets. Enumeration is a crucial skill in combinatorics, as it confirms the calculated possible outcomes.