Problem 3
Question
\(n / m\) means that \(n\) is a factor of \(m\), then the relation \(\%\) 'is (A) reflexive and symmetric. (B) transitive and reflexive. (C) reflexive, transitive and symmetric. (D) reflexive, transitive and not symmetric.
Step-by-Step Solution
Verified Answer
The relation is reflexive and transitive, but not symmetric. The answer is (D).
1Step 1: Understand the Relation Properties
To determine if a relation is reflexive, each element must relate to itself. For symmetry, if an element relates to another, the reverse must also be true. Transitivity implies that if one element relates to a second, which in turn relates to a third, the first must relate to the third.
2Step 2: Check Reflexivity
A relation is reflexive if every element divides itself. Given a positive integer \( n \), \( n/n = 1 \), so \( n \) is always a factor of itself. Hence, the relation is reflexive.
3Step 3: Check Symmetry
For symmetry, if \( n/m \), then \( m/n \) must also hold. However, this is not true as typically one number being a factor of another does not imply that the second is a factor of the first (e.g. 2 is a factor of 4, but not vice versa). Therefore, the relation is not symmetric.
4Step 4: Check Transitivity
For transitivity, if \( n/m \) and \( m/k \), then \( n/k \) should hold. This property holds because the product of factors is a multiple (e.g., if \( n \) divides \( m \) and \( m \) divides \( k \), then \( n \) divides \( k \)). Thus, the relation is transitive.
5Step 5: Choose the Correct Answer
Based on the analysis, the relation is reflexive and transitive but not symmetric. Thus, the correct option is (D) reflexive, transitive, and not symmetric.
Key Concepts
ReflexivitySymmetryTransitivity
Reflexivity
A relation is considered reflexive if every element in a set is related to itself. In practical terms, this means that for any element \( n \) within the set, the relation must satisfy \( n / n \), which equals 1. This is always true because any number divides itself.
For example, consider the set of positive integers. The reflexive property holds here because each integer is a factor of itself. This makes the relation of "being a factor" inherently reflexive. It's important to remember that reflexivity is a fundamental property of many mathematical relations, ensuring that all elements relate back to themselves in a predictable manner. This feature is particularly useful when evaluating or proving further properties of mathematical relations.
For example, consider the set of positive integers. The reflexive property holds here because each integer is a factor of itself. This makes the relation of "being a factor" inherently reflexive. It's important to remember that reflexivity is a fundamental property of many mathematical relations, ensuring that all elements relate back to themselves in a predictable manner. This feature is particularly useful when evaluating or proving further properties of mathematical relations.
Symmetry
Symmetry in a relation indicates that if one element is related to another, then the reverse relation must also be true. More formally, if \( n / m \) is true, then \( m / n \) must be true as well for the relation to be symmetric.
However, this property does not hold in the context of factors. For instance, while 2 is a factor of 4 (\( 2 / 4 \)), 4 is not a factor of 2 (\( 4 / 2 \)). Thus, this negates symmetry in the factor relation. Whenever testing for symmetry, it's crucial to check both directions of the relation. This ensures that the dependability of the relation to show equivalent results when the order of elements is reversed.
This lack of symmetry also highlights the importance of thoroughly verifying properties as assumptions about relations can lead to incorrect conclusions if not carefully examined.
However, this property does not hold in the context of factors. For instance, while 2 is a factor of 4 (\( 2 / 4 \)), 4 is not a factor of 2 (\( 4 / 2 \)). Thus, this negates symmetry in the factor relation. Whenever testing for symmetry, it's crucial to check both directions of the relation. This ensures that the dependability of the relation to show equivalent results when the order of elements is reversed.
This lack of symmetry also highlights the importance of thoroughly verifying properties as assumptions about relations can lead to incorrect conclusions if not carefully examined.
Transitivity
Transitivity in mathematical relations is a property that connects sequences of elements. Specifically, if an element \( n \) is related to \( m \), and \( m \) is related to \( k \), then \( n \) should also be related to \( k \).
When applied to the relation of factors, transitivity holds true. For example, assume \( n \) is a factor of \( m \) and \( m \) is a factor of \( k \). Then it logically follows that \( n \) is a factor of \( k \). This is because the multiplicative path formed by these relations ensures that divisibility is consistent throughout.
When applied to the relation of factors, transitivity holds true. For example, assume \( n \) is a factor of \( m \) and \( m \) is a factor of \( k \). Then it logically follows that \( n \) is a factor of \( k \). This is because the multiplicative path formed by these relations ensures that divisibility is consistent throughout.
- For instance, if \( 2 / 4 \) and \( 4 / 8 \), then by necessity, \( 2 / 8 \).
- This reveals the strength of transitive properties, as it links simple relations to build a coherent and consistent map of how elements interconnect.
Other exercises in this chapter
Problem 1
Let \(F_{1}\) be the set of all parallelograms, \(F_{2}\) the set of rectangles, \(F_{3}\) the set of rhombuses, \(F_{4}\) the set of squares and \(F_{5}\) the
View solution Problem 2
(i) Let \(R\) be the relation on the set \(R\) of all real numbers defined by setting \(a R b\) if \(|a-b| \leq \frac{1}{2}\). Then \(R\) is (A) Reflexive and s
View solution Problem 4
Set \(A\) and \(B\) have 3 and 6 elements respectively. What can be the minimum number of elements in \(A \cup B ?\) (A) 18 (B) 9 (C) 6 (D) 3
View solution Problem 5
Let \(R\) be a relation defined on the set of natural numbers \(N\) as \(R=[(x, y): x \in N, y \in N, 2 x+y=41]\). Then (A) Domain of \(R=\\{1,2,3, \ldots, 19,2
View solution