Relations

Question

The relation R on the set {1, 2, 3, ..., 10} defined by R = {(x, y) : x and y have exactly one common factor other than 1} is

RankGuru · Accepted Answer

1. **Test Reflexive**: For $(x, x)$ to be in $R$, $x$ must have exactly one factor other than 1. This would only be true for prime numbers. For $x=4$, factors are $\{1, 2, 4\}$ (two factors other than 1). Thus, not reflexive for all $x$.
2. **Test Symmetric**: If $(x, y)$ is a pair where they share exactly one prime factor, then surely $(y, x)$ shares that same property. **Symmetric** (Yes).
3. **Test Transitive**: 
   - Let $x=4, y=6$. Common factor is $\{2\}$. So $(4,6) \in R$.
   - Let $y=6, z=9$. Common factor is $\{3\}$. So $(6,9) \in R$.
   - For transitivity, $(4,9)$ must be in $R$. But $4, 9$ have **zero** common factors other than 1. Not transitive.
4. **Conclusion**: Symmetric only.

Relations

Select the correct option:

🔒 Solution Hidden from View

More sets, relations and functions Practice Questions

If R is a relation from a finite set A having m elements to a finite set B having n elements, then t...

Let R be a relation on the set N of natural numbers defined by nRm \u21Longleftrightarrow n is a fac...

The relation 'less than' on the set of real numbers R is

Let R be the relation on the set R of all real numbers defined by aRb \u21Longleftrightarrow |a - b|...

If A = {a, b, c}, then the number of equivalence relations on A containing (a, b) is

The minimum number of ordered pairs that must be added to the relation R = {(1, 2), (2, 3)} on the s...

Let R be a relation on the set N of natural numbers defined by R = {(x, y) : x + 2y = 8}. The range ...

A relation R on set A = {1, 2, 3, 4} is defined as R = {(1,1), (2,2), (3,3), (4,4), (1,2), (2,1)}. W...

The relation R = {(1,1), (2,2), (3,3)} on the set A = {1, 2, 3} is

Let R be a relation on Z (set of integers) defined by R = {(a, b) : a - b is divisible by 5}. Then R...

The number of reflexive relations on a set with 3 elements is

If R and S are two equivalence relations on a set A, then which of the following is always an equiva...

About This Question

Solution