Question

Let \(A=\\{a, b, c, d, e\\} .\) Suppose \(R\) is an equivalence relation on \(A .\) Suppose also that \(a R d\) and \(b R c, e R a\) and \(c R e .\) How many equivalence classes does \(R\) have?

Short answer

There are 2 equivalence classes in the given equivalence relation on \(A\).

Step-by-step solution

Analysis of the equivalence relations

Equivalence relations are described in this problem as follows: \(a R d, b R c, e R a\) and \(c R e\). We can see that \(a, d, e\) are related to each other and \(b, c\) are related to each other. Now, let's combine these relations.

Combining the relations

We can combine these relations into two separate classes. The first class will contain the elements that are related to each other either directly or indirectly, i.e., \(a, d, e\). Similarly, the second class will contain the elements \(b\) and \(c\).

Count the classes

We have grouped the elements of \(A\) into two distinct equivalence classes based on the relations provided. These classes are: \(\{a, d, e\}\) and \(\{b, c\}\).

Key concepts

Equivalence Relation

Understanding equivalence relations is crucial to set theory and various branches of mathematics. An equivalence relation on a set is a special type of relationship between the elements of the set that satisfies three main properties: reflexivity, symmetry, and transitivity.

• Reflexivity means each element is related to itself. In formal terms, for any element 'a' in set A, we have aRa.
• Symmetry stipulates that if one element is related to another, then that relationship is bidirectional. For instance, if we have aRb, then we must also have bRa.
• Transitivity indicates that if one element is related to a second, which is in turn related to a third, then the first element is related to the third. Formally, if aRb and bRc, then aRc.

These properties ensure an element can be classified into a specific group called an equivalence class. An equivalence class for an element 'a' in A is the set of all elements in A that are equivalent to 'a'.

In the exercise,

Set Theory

Set theory is the mathematical study of collections of objects, known as sets. It serves as the foundation for many areas of mathematics, where we define sets based on certain conditions and analyze relations between sets. A set does not consider the order of elements and does not contain duplicate elements.

Equivalence classes arise naturally in set theory. Each class is essentially a subset of a set that groups together elements which are equivalent under a specified equivalence relation, segmenting the set into non-overlapping parts.

When determining equivalence classes, we often visually represent them using Venn diagrams or sets notation to illustrate their distinctiveness within the universal set. For example, our exercise indicates that the set A contains 5 elements. Utilizing the given equivalence relation, we identify subgroups within A where the elements are equivalent to each other, forming the equivalence classes.

Mathematical Proof

A mathematical proof is a logical argument that establishes the truth of a mathematical statement beyond any doubt. It consists of a sequence of deductive steps that show why a statement must be true, using accepted mathematical principles and previously proven statements.

Structure of a Proof

When constructing a proof, mathematicians often begin with known information or postulates, apply logical reasoning, and use defined operations to reach a conclusion. Proofs can be direct, where the sequence of reasoning goes straight from assumptions to the conclusion, or indirect, where a supposition is proved by contradiction.

In the given exercise, the solution moves through a sequence of logical steps to show how the elements of set A relate to each other through the equivalence relation and then determine the number of equivalence classes in a methodical way. This step-by-step approach demonstrates the understated simplicity of mathematical proof - it's about connecting the dots one by one to arrive at the undeniable conclusion.