Problem 6 - A Gentle Introduction to the Art of Mathematics

Question

Consider the set \(\\{1,2,3, \ldots, 10\\}\). Express the characteristic function of the subset \(S=\\{1,2,3\\}\) as a set of ordered pairs.

Step-by-Step Solution

Answer

{(1, 1), (2, 1), (3, 1), (4, 0), (5, 0), (6, 0), (7, 0), (8, 0), (9, 0), (10, 0)}

Step 1 - Understand the Characteristic Function

The characteristic function of a subset, say \(S\), is a function that maps each element of the universal set to 1 if it belongs to \(S\) and to 0 if it does not.

Step 2 - Identify the Universal Set

The universal set in this problem is \( U = \{1,2,3, \ldots, 10\} \). This set contains all the elements from 1 to 10.

Step 3 - Identify the Subset

The subset given is \( S = \{1, 2, 3\} \). This subset contains the elements 1, 2, and 3.

Step 4 - Construct the Characteristic Function

For each element in the universal set \( U \), determine if it belongs to the subset \( S \). If it does, the characteristic function (as a set of ordered pairs) will include that element associated with 1; otherwise, with 0.

Step 5 - Create the Ordered Pairs

Following the rule from Step 4, we have:For 1: (1, 1)For 2: (2, 1)For 3: (3, 1)For 4: (4, 0)For 5: (5, 0)For 6: (6, 0)For 7: (7, 0)For 8: (8, 0)For 9: (9, 0)For 10: (10, 0)

Key Concepts

Subset

A subset is a set in which all its elements are also contained within another set. Let's denote this other set as the universal set, represented by the symbol \( U \). When we talk about subsets, we use the notation \( S \subseteq U \) to indicate that every element in \( S \) is also an element in \( U \).

For example, consider the subset \( S = \{1, 2, 3\} \) from the exercise. In our case:

• The universal set \( U = \{1, 2, 3, \, \ldots, \, 10\} \)
• The subset \( S \) consists of the elements 1, 2, and 3

Identifying which elements of the universal set belong to the subset helps us understand the relationships within the sets.

Universal Set

The universal set encompasses all the objects or elements under consideration in a particular discussion. In mathematical problems, the universal set is usually defined at the outset and contains every possible element that relates to the problem.

In this exercise, the universal set is:

\[ U = \{1, 2, 3, \ldots, 10\} \]

This means we are considering only the numbers from 1 to 10.

When working with subsets, it is important to clearly identify the universal set to define the scope within which subsets are examined. It is essentially the 'big picture' set containing all elements of interest.

Ordered Pairs

An ordered pair is a pair of elements in which the order matters. It is written in the form \((a, b)\), where \( a \) is the first element and \( b \) is the second element. Ordered pairs are particularly useful for functions and relations in mathematics.

In our exercise, we use ordered pairs to represent the characteristic function of the subset \( S \). The characteristic function maps each element of the universal set \( U \) to either 1 or 0, depending on whether the element is in the subset \( S \) or not. For example:

• For element 1, since it's in the subset \( S \), the ordered pair is \((1, 1)\).
• For element 10, since it's not in the subset \( S \), the ordered pair is \((10, 0)\).

Constructing such ordered pairs for each element in \( U \) results in the full characteristic function representation.