Question

The relation \(\mathrm{R}\) defined on the let \(\mathrm{A}=\\{1,2,3,4,5\\}\) by \(R=\left\\{(x, y) /\left|x^{2}-y^{2}\right|<16\right\\} \quad\) is given by (a) \(\\{(1,1),(2,1),(3,1),(4,1),(2,3)\\}\) (b) \(\\{(2,2),(3,2),(4,2),(2,4)\\}\) (c) \(\\{(3,3),(4,3),(5,4),(3,4)\\}\) (d) None of these

Short answer

The correct answer is \((d) \text{ None of these}\).

Step-by-step solution

Arrange Possible Pairs in Set A

Before testing the condition, let's write down all possible pairs in set A since it has only 5 elements. Remember that these relations are reflexive, so pairs like (1,1), (2,2), etc., should be considered. All possible pairs: {(1,1), (1,2), (1,3), (1,4), (1,5), (2,2), (2,3), (2,4), (2,5), (3,3), (3,4), (3,5), (4,4), (4,5), (5,5)}

Apply Condition to Each Pair and Find Valid Pairs

Now let's apply the condition |x^2 - y^2| < 16 to each of the pairs and determine whether they satisfy the condition: (1,1): |1^2 - 1^2| = 0 < 16; true (1,2): |1^2 - 2^2| = |-3| = 3 < 16; true (1,3): |1^2 - 3^2| = |-8| = 8 < 16; true (1,4): |1^2 - 4^2| = |-15| = 15 < 16; true (1,5): |1^2 - 5^2| = |-24| = 24 < 16; false (2,2): |2^2 - 2^2| = 0 < 16; true (2,3): |2^2 - 3^2| = |-5| = 5 < 16; true (2,4): |2^2 - 4^2| = |-12| = 12 < 16; true (2,5): |2^2 - 5^2| = |-21| = 21 < 16; false (3,3): |3^2 - 3^2| = 0 < 16; true (3,4): |3^2 - 4^2| = |-7| = 7 < 16; true (3,5): |3^2 - 5^2| = |-16| = 16 < 16; false (4,4): |4^2 - 4^2| = 0 < 16; true (4,5): |4^2 - 5^2| = |-9| = 9 < 16; true (5,5): |5^2 - 5^2| = 0 < 16; true Valid pairs: {(1,1), (1,2), (1,3), (1,4), (2,2), (2,3), (2,4), (3,3), (3,4), (4,4), (4,5), (5,5)}

Compare Result to Given Options

Let's now compare the resultant set of valid pairs to the options given in the exercise: (a) {(1,1),(2,1),(3,1),(4,1),(2,3)} != our valid pairs (b) {(2,2),(3,2),(4,2),(2,4)} != our valid pairs (c) {(3,3),(4,3),(5,4),(3,4)} != our valid pairs The set of valid pairs we found does not match any of the given options, so the correct answer is: (d) None of these

Key concepts

Set Theory

Set theory is a branch of mathematical logic that studies sets, which are collections of objects. In this exercise, the set \(A = \{1, 2, 3, 4, 5\}\)\ is used. Sets are essential because they allow us to group items in a structured way. The set \(A\) comprises natural numbers from 1 to 5.
When working with sets, we often look at relations between elements. A **relation** refers to a rule that links elements of one set with elements of another. This set \(A\) is being used to form pairs by examining a specific condition: \(R = \{(x, y) \ | \ |x^2 - y^2| < 16\}\).
In simpler words, this condition checks how the squares of these numbers relate to each other based on a particular inequality. When we analyze these relations, there are mathematical tools and conditions we use.

• Reflexive Relation: Each element relates to itself, such as (1,1), (2,2).
• Irreflexive Relation: No element is related to itself.
• Symmetric Relation: If (a,b) is in relation, then (b,a) is also.
• Transitive Relation: If (a,b) and (b,c) are in relation, then (a,c) must be.

Reflexive Relation

A reflexive relation is a type of relationship in which every element in a set is related to itself. This means, when you have a set \(A\), a reflexive relationship on \(A\) would include pairs like (1,1), (2,2), ..., (5,5). These can be thought of as the simplest form of pairing elements, where each "talks to" itself.
For the relation given in the exercise, to be reflexive, every element from the set \(A\) = \{1, 2, 3, 4, 5\} must have a pair like (x, x) which satisfies the condition \(|x^2 - x^2| < 16\). It's already guaranteed because \(0 < 16\).
However, for the given relation \(R\), not all the potential pairs exist, meaning \(R\) is not reflexive overall since not every element has its respective (x,x) pair maintained.

• Example of reflexive relation: \(\{ (1,1), (2,2), (3,3)\}\)
• Key point: Must include all self-pairs like (x, x).

Mathematical Conditions

Mathematical conditions are specific criteria that determine the relations between elements under consideration. These conditions often involve inequalities, as seen in our exercise, where the condition \( |x^2 - y^2| < 16\) determines the valid pairs.
This condition says that the absolute difference between the squares of any two numbers in the pair must be less than 16. Here’s how you can approach it:

• Calculate \(x^2\) and \(y^2\) for each element.
• Find the absolute difference: \(|x^2 - y^2|\).
• Check if this value is less than 16.

By verifying this, any pair (x, y) satisfying the criterion becomes part of the relation. It's crucial to follow this systematic approach for all elements to find the correct valid pairs.
Another essential aspect of such conditions is understanding their implications: a broader insight into the type of logic they're guiding us towards within a problem. The comparison not only tells us which pairs are included but also enhances our understanding of inequality usage in relations.