Problem 20 Is sameShirtColor symmetric?... [FREE SOLUTION]

Chapter 3: Problem 20

Is sameShirtColor symmetric?

Short Answer

Expert verified

The sameShirtColor relation is symmetric. This is because if any two students, say x and y, are wearing the same color shirt, they are related by the sameShirtColor relation (x, y). This also implies that y and x have the same shirt color, and hence, (y, x) also holds. Therefore, if (x, y) is in the relation, (y, x) is also in the relation, satisfying the definition of symmetry.

Step by step solution

Definition of Symmetric Relation

A relation R on a set A is called symmetric if for all (x, y) in R, (y, x) is also in R. In simple language, if "x is related to y", then "y is related to x" for the sameShirtColor relation.

Analyzing the sameShirtColor relation

Consider two students x and y. Let's analyze the conditions for their relation with respect to sameShirtColor. If students x and y have the same shirt color, they are related by sameShirtColor relation. In this case, the relation will be (x, y). Now, if students y and x have the same shirt color (which is true, since they are wearing the same color per our initial condition), the relation will become (y, x).

Conclusion

Since both relations (x, y) and (y, x) are present whenever students x and y have the same shirt color, we can conclude that the sameShirtColor relation is symmetric.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equivalence Relation

The concept of an equivalence relation is an important one in mathematics. It provides a framework to define a certain kind of relationship between elements of a set that follows three specific rules: reflexivity, symmetry, and transitivity. In order to be called an equivalence relation, a relation must satisfy all three of these properties.

Let's break down each one with a simple example:

Reflexivity: Every element must relate to itself. For instance, if you have a shirt color relation, saying any student's shirt color is the same as itself is reflexively true.
Symmetry: If one element is related to another, then the second is related to the first. In our shirt color example, if Student A is wearing the same shirt color as Student B, then Student B is also wearing the same shirt color as Student A.
Transitivity: If one element is related to a second, and the second is related to a third, then the first must be related to the third. Taking shirts again, if Student A's shirt matches Student B's, and Student B's shirt matches Student C's, then Student A's shirt must match Student C's shirt.

Understanding these properties is key to identifying and discussing equivalence relations in mathematics and other fields.

Discrete Mathematics

Discrete Mathematics is a branch of mathematics dealing with discrete elements that uses algebra and arithmetic. It is essential in computer science and information technology because it includes topics that are at the core of these fields.

Discrete Mathematics covers various topics, such as:

Set Theory: The study of sets, or collections of objects. It provides a foundation for various other topics by helping define and distinguish concepts.
Logic: The formal logical reasoning that provides the framework for proving mathematical statements and constructing algorithms.
Combinatorics: The study of counting, arrangement and combination of elements. This is vital for understanding possibilities and probabilities, especially in the field of computer science for permutations and different arrangements.
Graph Theory: The study of graphs and networks, which has applications in routing, networking, and data organization.

These diverse topics provide tools used in computer algorithms, cryptography, and network design. Understanding the principles of Discrete Mathematics enables us to design better systems and solve complex computational problems.

Relation Properties

To understand relations deeply, it's vital to explore their properties. Relations between elements in sets can exhibit several characteristics like reflexivity, symmetry, and transitivity, as mentioned earlier.

Beyond these, relations can also be classified by other properties:

Antisymmetry: In some relations, if (a, b) is present and also (b, a), then a must equal b. This is often seen in ordering systems, like less than or equal relations.
Irreflexivity: This is the opposite of reflexivity. A relation is irreflexive if no element is related to itself. An example is the relation "is a parent of," which clearly cannot occur between one person and themselves.
Asymmetry: A relation is asymmetric if it does not have the symmetry property at all; that is, if (x, y) is in the relation, then (y, x) must not be. For example, the relation "is older than" is asymmetric.
Transitivity: Similar to what makes a relation an equivalence, transitivity in more generalized relations ensures that relatedness maintains a chain-like progression.

Understanding these properties helps one classifying and leveraging relations to solve problems across mathematics and applied sciences. Knowing how these properties manifest in different contexts is fundamental to grasping the complexity and utility of relations.

91影视

Is sameShirtColor symmetric?

Short Answer

Step by step solution

Definition of Symmetric Relation

Analyzing the sameShirtColor relation

Conclusion

Key Concepts

Equivalence Relation

Discrete Mathematics

Relation Properties

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