Question

Let \(A=\\{1,2,3,4\\}\) and \(B=\\{a, b, c\\} .\) Define a function \(F: A \rightarrow B\) as \(F(1)=a\). \(F(2)=b, F(3)=c,\) and \(F(4)=c .\) List the ordered pairs of the equivalence relation \(R\) defined on \(A\) as \(x R y\) if and only if \(F(x)=F(y)\). List the elements of the partition of A determined by this equivalence relation.

Short answer

Ordered pairs: \(\{(1,1), (2,2), (3,3), (3,4), (4,3), (4,4)\}\). Partition: \(\{\{1\}, \{2\}, \{3,4\}\}\).

Step-by-step solution

Understand the Problem

We need to find the ordered pairs \(x, y\) such that \(x \, R \, y\) according to the equivalence relation \(R\), which is defined as \(F(x) = F(y)\). We then need to find the partition of the set \(A\) determined by this equivalence relation.

Check Each Pair in A against F

For relation \(R\), we compare every element in \(A\) to check where \(F(x) = F(y)\). - \(F(1) = a\).- \(F(2) = b\).- \(F(3) = c\), and \(F(4) = c\).This means, \(x \, R \, y\) if \(x \, ext{and} \, y\) have the same outputs with function \(F\).

Determine Ordered Pairs

Based on \(F\), we evaluate: - Elements for which \(F(x) = F(y) = a\): \( (1, 1) \)- Elements for which \(F(x) = F(y) = b\): \( (2, 2) \)- Elements for which \(F(x) = F(y) = c\): \( (3, 3), (3, 4), (4, 3), (4, 4) \)Thus the ordered pairs are \(\{(1, 1), (2, 2), (3, 3), (3, 4), (4, 3), (4, 4)\}\).

Determine the Partition of A

The partition of \(A\) involves grouping elements with the same function result into subsets:- \(\{1\}\) since \(F(1) = a\). - \(\{2\}\) since \(F(2) = b\). - \(\{3, 4\}\) since \(F(3) = F(4) = c\).

Key concepts

Function Mapping

In the world of mathematics, functions play a pivotal role in relating elements from one set to another. A function mapping establishes a relationship where each input from a set, referred to as the domain, is paired with exactly one output in another set, known as the codomain. This relationship is crucial in understanding how elements interact with each other.
For example, in our given problem, the function \(F\) maps elements of set \(A = \{1, 2, 3, 4\}\) to set \(B = \{a, b, c\}\). The function is defined such that:
- \(F(1) = a\)
- \(F(2) = b\)
- \(F(3) = c\)
- \(F(4) = c\)
This means that each element in \(A\) is given a specific output in \(B\). When defining a function, it ensures that for any input \(x\) in the domain \(A\), there is a unique output \(y\) in the codomain \(B\).

Partition of a Set

The concept of partitioning a set involves dividing it into non-overlapping subsets, where each element of the original set belongs to exactly one subset. This is a core concept in discrete mathematics, often encountered when dealing with equivalence relations.
In the exercise, the equivalence relation \(R\) defined by \(F(x) = F(y)\) helps to establish the partitions of set \(A\). The elements that map to the same result in set \(B\) are grouped together:
- \(\{1\}\) forms a subset because \(F(1) = a\).
- \(\{2\}\) forms another distinct subset as \(F(2) = b\).
- \(\{3, 4\}\) belong to the same subset since both \(F(3)\) and \(F(4)\) yield \(c\).
Thus, the entire set \(A\) is partitioned into \(\{\{1\}, \{2\}, \{3, 4\}\}\), where each subset contains elements having the same function result.

Discrete Mathematics

Discrete mathematics is the study of mathematical structures that are fundamentally discrete, not supporting or requiring the notion of continuity. It encompasses a variety of topics such as logic, set theory, graph theory, and combinatorics.
When dealing with equivalence relations and partitions, we're diving into a domain that is deeply rooted in discrete mathematics principles. Equivalence relations, such as the one described in the exercise, demonstrate how elements in a set can be grouped based on shared properties, which in turn leads to the formation of partitions.
Understanding discrete mathematics is pivotal for solving problems in computer science, cryptography, and network theory, among others. It lays the groundwork for algorithms and structures that help us process and analyze discrete systems. The exercise we tackled here shows a practical application of these principles, illustrating how discrete mathematics helps organize and interpret complex relationships within sets.