Exploring Relations: Matrices, Graphs, And Properties

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Hey guys! Let's dive into the fascinating world of relations in mathematics. We're going to explore this using a specific example and break down the concepts in a way that's easy to understand. We'll be looking at how to represent relations using matrices and directed graphs, and we'll also investigate the key properties of these relations. This is a super important topic, so pay close attention because we're going to break down some complex ideas into easy-to-digest pieces. Ready to get started? Let's go!

(a) Determine the Matrix M_R of R

Alright, first things first. We've got a set, let's call it A, which is defined as A = {1, 2, 3, 4, 5, 6}. Now, we have a relation, labeled R, defined on A. This relation dictates that 'a' is related to 'b' (written as aRb) if and only if a - b ≥ 2. Basically, this means that the difference between 'a' and 'b' has to be greater than or equal to 2 for them to be related. Got it? For example, if we take the numbers 6 and 3, since 6 - 3 = 3 (which is greater than or equal to 2), then 6 is related to 3, or (6, 3) ∈ R. But if we try 2 and 3, since 2 - 3 = -1 (which is NOT greater than or equal to 2), then 2 is not related to 3, or (2, 3) ∉ R. Makes sense, right?

Now, how do we represent this relation in a matrix? This matrix, M_R, is a way to visually show which elements of A are related to each other. Because our set A has 6 elements, our matrix M_R will be a 6x6 matrix. The rows and columns will both correspond to the elements of A (1, 2, 3, 4, 5, and 6). We'll fill in the matrix with 0s and 1s. A '1' in the (i, j) position means that the element in the i-th row is related to the element in the j-th column. A '0' means they are not related. So, let's build the matrix M_R. This is where the fun starts!

Let's go through it systematically. For the element 1, can it be related to any other element? No, because there's no number in A that's less than or equal to -1. So, the first row will have all zeros: [0, 0, 0, 0, 0, 0]. For the element 2, it can be related to nothing. Thus, second row will also be [0, 0, 0, 0, 0, 0]. For the element 3, it can only be related to 1, since 3 - 1 = 2 ≥ 2. So the row is [1, 0, 0, 0, 0, 0]. Continuing this way, we have:

  • For 4: 4 - 1 = 3 ≥ 2 and 4 - 2 = 2 ≥ 2. So we get [1, 1, 0, 0, 0, 0].
  • For 5: 5 - 1 = 4 ≥ 2, 5 - 2 = 3 ≥ 2, and 5 - 3 = 2 ≥ 2. Hence, [1, 1, 1, 0, 0, 0].
  • For 6: 6 - 1 = 5 ≥ 2, 6 - 2 = 4 ≥ 2, 6 - 3 = 3 ≥ 2, and 6 - 4 = 2 ≥ 2. Therefore, [1, 1, 1, 1, 0, 0].

So, the matrix M_R looks like this:

0 0 0 0 0 0
0 0 0 0 0 0
1 0 0 0 0 0
1 1 0 0 0 0
1 1 1 0 0 0
1 1 1 1 0 0

That's our matrix M_R! Now you can easily see which elements are related to each other at a glance. We can also see how sparse it is, which tells us that our relation R is pretty restrictive. Only a few pairs of numbers satisfy the condition a - b ≥ 2. Understanding the construction of the matrix is key to understanding how relations work. Always remember to start from the basis, and you can solve the problem easily.

(b) Draw the Directed Graph of (A, R)

Okay, now let's visualize our relation R using a directed graph. A directed graph, also known as a digraph, is a visual representation of a relation. It consists of nodes (which represent the elements of our set A) and directed edges (arrows) that show the relationships between those elements. An arrow goes from 'a' to 'b' if aRb. Let's see how we can draw this graph.

We start by drawing six nodes, one for each element in our set A: 1, 2, 3, 4, 5, and 6. Now, for each pair (a, b) in the relation R, we draw an arrow from node 'a' to node 'b'. Looking back at our relation, we know that:

  • (3, 1) ∈ R because 3 - 1 = 2 ≥ 2. So, we draw an arrow from 3 to 1.
  • (4, 1) ∈ R because 4 - 1 = 3 ≥ 2, and (4, 2) ∈ R because 4 - 2 = 2 ≥ 2. So, we draw arrows from 4 to 1 and from 4 to 2.
  • (5, 1) ∈ R, (5, 2) ∈ R, and (5, 3) ∈ R. We draw arrows from 5 to 1, 5 to 2, and 5 to 3.
  • (6, 1) ∈ R, (6, 2) ∈ R, (6, 3) ∈ R, and (6, 4) ∈ R. We draw arrows from 6 to 1, 6 to 2, 6 to 3, and 6 to 4.

That's it! Once you have the nodes and draw the arrows according to the relation, you will obtain the directed graph. With this directed graph, you can now visually see which elements in A are related to each other according to the relation R. The graph makes it super easy to understand the relationships. It's like a map of the connections between the elements. This visual representation is great for grasping the nature of the relation and its properties.

Now, try drawing it yourself! Start with the nodes and then add the arrows based on the matrix or the original definition of the relation. Visualization is a key component to fully understand this mathematical concept. The diagram will bring the relation to life and help you see the relationships in a whole new way.

(c) Give the Properties of R (reflexive, symmetric, transitive, antisymmetric)

Alright, let's get into some important properties of our relation R. These properties help us classify and understand the behavior of the relation. We're going to check if R is reflexive, symmetric, transitive, and antisymmetric. Understanding these properties is crucial to fully grasp the characteristics of the relation. Let's break each of them down, one by one.

Reflexive

A relation R on a set A is reflexive if for every element 'a' in A, (a, a) ∈ R. In simpler terms, an element must be related to itself. Let's check our relation. We need to see if a - a ≥ 2 for all 'a' in A. Since a - a = 0, and 0 is not greater than or equal to 2, no element is related to itself. Thus, the relation R is not reflexive. Easy enough, right?

Symmetric

A relation R on a set A is symmetric if, for all a, b ∈ A, whenever (a, b) ∈ R, then (b, a) ∈ R. In other words, if 'a' is related to 'b', then 'b' must also be related to 'a'. Let's check this for our relation. For (a, b) ∈ R, we have a - b ≥ 2. This does not imply that b - a ≥ 2. For instance, (6, 3) ∈ R because 6 - 3 = 3 ≥ 2. But (3, 6) ∉ R because 3 - 6 = -3, which is not ≥ 2. So, the relation R is not symmetric.

Transitive

A relation R on a set A is transitive if, for all a, b, c ∈ A, whenever (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R. If 'a' is related to 'b', and 'b' is related to 'c', then 'a' must be related to 'c'. Looking at our relation, if a - b ≥ 2 and b - c ≥ 2, we would need to have a - c ≥ 2. However, consider an example. Let's say (6, 3) ∈ R and (3, 1) ∈ R. If the relation was transitive, we would also need to have (6, 1) ∈ R. But since 6 - 1 = 5 ≥ 2, so (6, 1) ∈ R. This appears to satisfy the requirement, but to be truly transitive, all cases must hold true. By careful inspection of the set and relation, since we cannot find a case where (a, b) ∈ R and (b, c) ∈ R but (a, c) ∉ R, then this implies that the relation R is transitive. So, our relation is transitive.

Antisymmetric

A relation R on a set A is antisymmetric if, for all a, b ∈ A, whenever (a, b) ∈ R and (b, a) ∈ R, then a = b. This is a bit more subtle. It essentially means that if 'a' is related to 'b' and 'b' is related to 'a', then 'a' and 'b' must be the same element. Let's revisit our definition. We already know that our relation R is not symmetric. Therefore, there are no pairs (a, b) and (b, a) where both are in R, except the case when a = b. Thus, the relation R is antisymmetric. Remember that if a relation isn't symmetric, it can still be antisymmetric.

In Summary, for the set A and relation R, we have:

  • Not Reflexive
  • Not Symmetric
  • Transitive
  • Antisymmetric

And that wraps it up! You've now got a good handle on how to analyze relations and understand their properties. Keep practicing with different examples, and you'll become a pro in no time! Keep up the great work, and don't hesitate to ask if you have any questions!