Let \( R \) be a relation on \( \mathbb{Z} \times \mathbb{Z} \) defined by \((a, b) R (c, d)\) if and only if \(ad - bc\) is divisible by 5.
Then \( R \) is:
Then \( R \) is:
- Reflexive but neither symmetric nor transitive
- Reflexive and symmetric but not transitive
- Reflexive, symmetric and transitive
- Reflexive and transitive but not symmetric
The Correct Option is B
Approach Solution - 1
To determine if the given relation \( R \) on \( \mathbb{Z} \times \mathbb{Z} \) is reflexive, symmetric, and/or transitive, we analyze each property in the context of the definition of \( R \): \((a, b) R (c, d)\) if and only if \(ad - bc\) is divisible by 5.
- Reflexivity:
- A relation \( R \) is reflexive if \((a, b) R (a, b)\) for all \((a, b) \in \mathbb{Z} \times \mathbb{Z}\).
- Substitute \((a, b)\) for \((c, d)\): we get \(ad - bc = ab - ba = 0\).
- Since 0 is divisible by 5, the relation is reflexive.
- Symmetry:
- A relation \( R \) is symmetric if whenever \((a, b) R (c, d)\), then \((c, d) R (a, b)\).
- If \(ad - bc\) is divisible by 5, \(-ad + bc\) is also divisible by 5 (because \(-1 \times\) a number divisible by 5 is also divisible by 5).
- Hence, the relation is symmetric.
- Transitivity:
- A relation \( R \) is transitive if whenever \((a, b) R (c, d)\) and \((c, d) R (e, f)\), then \((a, b) R (e, f)\).
- Let \(x = ad - bc\) and \(y = cf - de\), both divisible by 5. For transitivity, \(af - be\) should be divisible by 5.
- However, knowing \(x\) and \(y\) are divisible by 5 doesn't ensure \(af - be\) will also be divisible by 5.
- For example, if \(a = 1, b = 0, c = 0, d = 1, e = 1, f = 0\), then \(ad - bc = 1\), \(cf - de = 1\), but \(af - be = 0\), which is divisible by 5.
- Instead, if \( b = 1 \) and other choices remain 0, then it's not guaranteed that \(af - be = 0 - 1\) is not divisible by 5, specifically.
- Thus, the relation may not be transitive.
Considering these observations, the correct answer is: Reflexive and symmetric but not transitive.
Approach Solution -2
Reflexive : for \((a, b) R (a, b) \)
\( ⇒ ab – ab = 0\) is divisible by 5.
So \((a, b) R(a, b) ∀ a, b ∈ Z \)
∴ R is reflexive Symmetric : For \((a, b) R(c, d) \)
If \(ad – bc\) is divisible by 5.
Then \(bc – ad\) is also divisible by 5.
\(⇒ (c, d) R(a, b) ∀ a, b, c, d ∈ Z \)
∴ R is symmetric Transitive : If \((a, b) R(c, d) \)
\(⇒ ad – bc\) divisible by 5 and \((c, d) R (e, f) \)
\(⇒ cf – de\) divisible by 5
\(ad – bc = 5k_1\) \( k_1\) and \(k_2\) are integers
\(cf – de = 5k_2\)
\(afd – bcf = 5k_1f \)
\(bcf – bde = 5k_2b \)
\(afd – bde = 5(k_1f + k_2b) \)
\(d(af – be) = 5 (k_1f + k_2b) \)
\(⇒ af – be\) is not divisible by 5 for every a, b, c, d, e, f ∈ Z.
\( ∴\) R is not transitive
For e.g., take \(a = 1, b = 2, c = 5, d = 5, e = 2, f = 2\)
The correct option is (B): Reflexive and symmetric but not transitive
Top JEE Main Mathematics Questions
- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
- If , then the general value of is:
- The general solution of
- If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
Top JEE Main Relations Questions
- Let S = {1,2,3,..., 20}, R1 = {(a, b): a divide b}, R2 = {(a, b): a is integral multiple of b} and a, b ∈ S. n(R1 - R2) = ?
- A = {1, 2, 3, 4} minimum number of elements added to make an equivalence relation on set A containing (1, 3) & (1, 2) in it.
- If \(A = {1, 2, 3, … 100}\), \(R = {(x, y) | 2x = 3y, x, y ∈ A}\) is symmetric relation on \(A\) and the number of elements in \(R\) is \(n\), the smallest integer value of \(n\) is
- A = {1, 2, 3, 4} , R = {(1, 2), (2, 3), (2, 4)} R ⊆ S and S is an equivalence relation then the minimum number of elements to be added to R is n, then the value of n is?
- Define a relation \( R \) on the interval \( [0, \frac{\pi}{2}] \) by \( xRy \) if and only if \( \sec^2 x - \tan^2 y = 1 \). Then \( R \) is:
Top JEE Main Questions
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
A black body is at a temperature of 2880 K. The energy of radiation emitted by this body with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien's constant, b = 2.88×106 nm-K. Then,
Concepts Used:
Relations
A relation in mathematics defines the relationship between two different sets of information. If two sets are considered, the relation between them will be established if there is a connection between the elements of two or more non-empty sets. Therefore, we can say, ‘A set of ordered pairs is defined as a relation.’
Read Also: Relation and Function
Types of Relations:
There are 8 main types of relations which are:
- Empty Relation - An empty relation is one in which there is no relation between any elements of a set.
- Universal Relation - A universal is a type of relation in which every element of a set is related to each other. Now one of the universal relations will be R = {x, y} where, |x – y| ≥ 0. For universal relation, R = A × A
- Identity Relation - In an identity relation, every element of a set is related to itself only. For example, in a set A = {a, b, c}, the identity relation will be I = {a, a}, {b, b}, {c, c}.
- Inverse Relation - It is seen when a set has elements which are inverse pairs of another set. For example if set A = {(a, b), (c, d)}, then inverse relation will be R-1 = {(b, a), (d, c)}.
- Reflexive Relation - If every element of set A maps for itself, then set A is known as a reflexive relation.It is represented as a∈ A, (a,a) ∈ R.
- Symmetric Relation - A relation R on a set A is known as asymmetric relation if (a, b) ∈R then (b, a) ∈R , such that for all a and b ∈A.
- Transitive Relation - For transitive relation, if (x, y) ∈ R, (y, z) ∈ R, then (x, z) ∈ R. For a transitive relation, aRb and bRc ⇒ aRc ∀ a, b, c ∈ A
- Equivalence Relation - If a relation is reflexive, symmetric and transitive at the same time it is known as an equivalence relation.
Representation of Relations:
There are two ways by which a relation can be represented-
- Roster method
- Set-builder method
The roster form and set-builder for for a set integers lying between -2 and 3 will be-
Roster form
I= {-1,0,1,2}
Set-builder form
I= {x:x∈I,-2<x<3}