Question

Let \(X=\\{0,1\\}\). Let \(B=\mathcal{P}(X \times X)\) be the set of all binary relations on \(X\). (a) List all the elements of \(B\). (b) Since elements of \(B\) are themselves relations, it makes sense to ask whether two of those relations are inverses of each other. Let $$\text { IslnerseOf }=\left\\{(R, S): R \in B \text { and } S \in B \text { and } R=S^{-1}\right\\}$$ List all elements of IslmerseOf. (c) Since IslmerseOf is a binary relation, it has an inverse. What is IsimerseOf \(^{-1}\) ? (d) What is Islmerse of o IslnverseOf?

Short answer

(a) 16 elements in B. (b) 4 pairs in IslnerseOf. (c) IslnerseOf is self-inverse. (d) Identity relation.

Step-by-step solution

Define Elements of X

The set \(X\) is given as \(\{0, 1\}\). We will first list all the ordered pairs that can be formed from \(X \times X\): \[(0,0), (0,1), (1,0), (1,1)\].

Define the Power Set B

The binary relations on \(X\) are given by the power set of \(X \times X\), denoted by \(\mathcal{P}(X \times X)\). This means we need to find the power set of the four ordered pairs listed in Step 1.The power set \(B\) is: \[\emptyset, \{(0,0)\}, \{(0,1)\}, \{(1,0)\}, \{(1,1)\}, \{(0,0),(0,1)\}, \ldots, \{(0,0),(0,1),(1,0),(1,1)\}\]. There are \(2^4 = 16\) elements.

List Elements of Binary Relations

The set \(B = \mathcal{P}(X \times X)\) contains all the subsets of the set of pairs \((0,0), (0,1), (1,0), (1,1)\). We will list the elements:\[\begin{align*}B = \{&\emptyset, \{(0,0)\}, \{(0,1)\}, \{(1,0)\}, \{(1,1)\}, \{(0,0),(0,1)\}, \{(0,0),(1,0)\}, \{(0,0),(1,1)\}, \&\{(0,1),(1,0)\}, \{(0,1),(1,1)\}, \{(1,0),(1,1)\}, \{(0,0),(0,1),(1,0)\}, \&\{(0,0),(0,1),(1,1)\}, \{(0,0),(1,0),(1,1)\}, \{(0,1),(1,0),(1,1)\}, \&\{(0,0),(0,1),(1,0),(1,1)\}\}.\end{align*}\] There are 16 elements.

Define Inverse Relations

For a relation \(R\), the inverse \(R^{-1}\) swaps the elements of the pairs, i.e., if \((a,b) \in R\), then \((b,a) \in R^{-1}\). We check each pair in \(B\) for this property.

List IslnerseOf

We list pairs \((R, S)\) where \(R, S \in B\) and \(R = S^{-1}\). After checking the elements of \(B\): The set IslnerseOf contains these pairs:\[\begin{align*}&\{(\emptyset, \emptyset), \{\{(0,0)\}, \{(0,0)\}\}, \{\{(1,1)\}, \{(1,1)\}\}, \{\{(0,1), (1,0)\}, \{(0,1), (1,0)\}\}\}\end{align*}\]These are the only relations that are their own inverses.

Find IslnerseOf Inverse

The inverse of IslnerseOf will switch each \((R, S)\) to \((S, R)\). However, since each \(R = S^{-1}\) and vice versa in those pairs, IslnerseOf is its own inverse. Hence, \[\text{IslnerseOf}^{-1} = \text{IslnerseOf}\].

Find IslnerseOf o IslnerseOf

The composition \(\text{IslnerseOf} \circ \text{IslnerseOf}\) involves taking each \((R, S)\) from IslnerseOf and mapping \(S\) to another relation that makes it \(S^{-1}\). Given IslnerseOf is its inverse, the composition is the identity relation, meaning it maps each relation to the relation itself:\[\text{IslnerseOf} \circ \text{IslnerseOf} = \{(R, R) : R \in B\}\].

Key concepts

Binary Relations

Binary relations are a fundamental concept in discrete mathematics, essential for understanding the way elements in one set can relate to elements in another. In the context of set theory, a binary relation on a set \(X\) is typically a subset of the Cartesian product \(X \times X\).
This means each relation consists of ordered pairs taken from the sets involved.

• A pair \((a, b)\) is considered part of a binary relation \(R\) if \(a\) is related to \(b\) under \(R\).
• Binary relations can exhibit many properties, such as being reflexive (every element relates to itself), symmetric (if \(a\) is related to \(b\), then \(b\) is related to \(a\)), antisymmetric, or transitive.

Understanding binary relations helps to model connections in data, such as the relationship between different nodes in a graph.
It's a vital part of the study of relations in mathematics and computer science.

Inverse Relations

Inverse relations are fascinating because they tell us what happens when we reverse the ordering of pairs in a relation. For a given relation \(R\), its inverse \(R^{-1}\) contains all the flipped pairs from \(R\).
Let's say \(R\) includes the pair \((a, b)\); then \(R^{-1}\) will contain \((b, a)\). Each relation has a unique inverse, and some relations are equal to their inverses.

• If a relation is symmetric, then it will be equal to its inverse since reversing elements does not change the pairs.
• Inverse relations are especially useful when assessing certain properties, such as testing for symmetric or antisymmetric properties.

In discrete mathematics, understanding inverse relations is crucial for exploring and constructing other set properties in computational and mathematical settings.

Power Set

The power set is a set that contains all the subsets of a given set, including the empty set and the set itself. It's an important way to explore all possible combinations of elements within a set.
In our exercise, the power set \(B\) of \(X \times X\) provides all binary relations on \(X\). This is because each subset of \(X \times X\) represents a different possible relation.

• The number of subsets, and thus the number of elements in the power set, is \(2^n\), where \(n\) is the number of elements in the original set.
• For set \(X = \{0, 1\}\), the Cartesian product \(X \times X = \{(0,0), (0,1), (1,0), (1,1)\}\) has four pairs. Hence, the power set has \(2^4 = 16\) different subsets.

The power set concept allows mathematicians to discuss the universe of all possible configurations and relationships a set might possess.

Set Theory

Set theory is the mathematical study of collections of objects, known as sets. It's foundational for modern mathematics, providing a framework to discuss groups of numbers, functions, or other mathematical entities.
Some key principles in set theory include:

• Union and Intersection: Union combines all elements in the involved sets, while intersection takes only the common elements.
• Subset: One set is a subset of another if all elements of the first set are also elements of the second.
• Complement: The complement of a set includes all the elements not in that set, typically within a universal set context.

By utilizing these basics, along with concepts like the power set and Cartesian products, set theory provides a robust language to handle mathematical and real-world problems that involve grouping, partitioning, and other operations on collections. Being well-versed in set theory equips students with the tools for rigorous thinking and application across various domains of mathematics.