Let R = (3, 3), (6, 6), (9, 9), (12,...

MathematicsTypes of RelationsJEE Main 2005Easy

View in:English Hindi

Let $R = {(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)}$ be a relation on the set $A = {3, 6, 9, 12}$ . The relation is

(A)

reflexive and transitive only

(B)

reflexive only

(C)

an equivalence relation

(D)

reflexive and symmetric only

Answer:A

Visualized Solution (English)

Defining Set .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A and Relation .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } R

Set $A = {3, 6, 9, 12}$
Relation $R = {(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)}$

Checking Reflexivity

A relation is reflexive if $(a, a) \in R$ for all $a \in A$ .
Check elements: $(3, 3) \in R$ , $(6, 6) \in R$ , $(9, 9) \in R$ , $(12, 12) \in R$ .
Conclusion: The relation is reflexive.

Checking Symmetry

A relation is symmetric if $(a, b) \in R ⟹ (b, a) \in R$ .
Counter-example: $(6, 12) \in R$ but $(12, 6) \in / R$ .
Conclusion: The relation is not symmetric.

Checking Transitivity

A relation is transitive if $(a, b) \in R$ and $(b, c) \in R ⟹ (a, c) \in R$ .
Example: $(3, 6) \in R$ and $(6, 12) \in R ⟹ (3, 12) \in R$ .
Example: $(3, 3) \in R$ and $(3, 6) \in R ⟹ (3, 6) \in R$ .
Conclusion: The relation is transitive.

Final Conclusion

The relation is Reflexive and Transitive.
It is not Symmetric.
Final Answer: Reflexive and Transitive only.

00:00 / 00:00

Conceptually Similar Problems

MathematicsTypes of RelationsJEE Main 2004Easy

View in:English Hindi

Let $R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)}$ be a relation on the set $A = {1, 2, 3, 4}$ . The relation R is

(A)

reflexive

(B)

transitive

(C)

not symmetric

(D)

a function

Answer:C

MathematicsTypes of RelationsJEE Main 2006Easy

View in:English Hindi

Let W denote the words in the English dictionary. Define the relation R by $R = {(x, y) \in W \times W ∣ the words x and y have at least one letter in common.}$ Then R is

(A)

not reflexive, symmetric and transitive

(B)

reflexive, symmetric and not transitive

(C)

reflexive, symmetric and transitive

(D)

reflexive, not symmetric and transitive

Answer:B

MathematicsTypes of RelationsJEE Main 2010Easy

View in:English Hindi

Consider the following relations: R = {(x, y) | x, y are real numbers and x = wy for some rational number w}; S = {( $\frac{m}{n}, \frac{p}{q}$ ) | m, n, p and q are integers such that $n, q \neq = 0$ and qm = pn}. Then

(A)

Neither R nor S is an equivalence relation

(B)

S is an equivalence relation but R is not an equivalence relation

(C)

R and S both are equivalence relations

(D)

R is an equivalence relation but S is not an equivalence relation

Answer:B

MathematicsTypes of RelationsJEE Main 2011Easy

View in:English Hindi

Let R be the set of real numbers. Statement-1: $A = {(x, y) \in R \times R : y - x is an integer}$ is an equivalence relation on R. Statement-2: $B = {(x, y) \in R \times R : x = α y for some rational number α}$ is an equivalence relation on R.

(A)

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

(B)

Statement-1 is true, Statement-2 is false.

(C)

Statement-1 is false, Statement-2 is true.

(D)

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

Answer:B

MathematicsTypes of RelationsJEE Main 2008Easy

View in:English Hindi

Let $R$ be the real line. Consider the following subsets of the plane $R \times R$ : $S = {(x, y) : y = x + 1 and 0 < x < 2}$ , $T = {(x, y) : x - y is an integer}$ . Which one of the following is true?

(A)

Neither S nor T is an equivalence relation on R

(B)

Both S and T are equivalence relation on R

(C)

S is an equivalence relation on R but T is not

(D)

T is an equivalence relation on R but S is not

Answer:D

MathematicsClassification of FunctionsJEE Advanced 1979Easy

View in:English Hindi

Let $R$ be the set of real numbers. If $f : R \to R$ is a function defined by $f (x) = x^{2}$ , then $f$ is :

(A)

Injective but not surjective

(B)

Surjective but not injective

(C)

Bijective

(D)

None of these

Answer:D

MathematicsTypes of MatricesJEE Main 2011Easy

View in:English Hindi

Let $A$ and $B$ be two symmetric matrices of order 3. Statement-1: $A (B A)$ and $(A B) A$ are symmetric matrices. Statement-2: $A B$ is symmetric matrix if matrix multiplication of $A$ with $B$ is commutative.

(A)

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

(B)

Statement-1 is true, Statement-2 is false.

(C)

Statement-1 is false, Statement-2 is true.

(D)

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

Answer:A

MathematicsAlgebraic Operations on MatricesJEE Main 2006Easy

View in:English Hindi

Let $A = [1324]$ and $B = [a 0 0 b]$ , $a, b \in N$ . Then

(A)

there cannot exist any

B

such that

A B = B A

(B)

there exist more than one but finite number of

B

's such that

A B = B A

(C)

there exists exactly one

B

such that

A B = B A

(D)

there exist infinitely many

B

's such that

A B = B A

Answer:D

MathematicsClassification of FunctionsJEE Advanced 2012Moderate

View in:English Hindi

The function $f : [0, 3] \to [1, 29]$ , defined by $f (x) = 2 x^{3} - 15 x^{2} + 36 x + 1$ , is

(A)

one-one and onto

(B)

onto but not one-one

(C)

one-one but not onto

(D)

neither one-one nor onto

Answer:B

MathematicsDistance and Section FormulasJEE Main 2002Easy

View in:English Hindi

A triangle with vertices $(4, 0), (- 1, - 1), (3, 5)$ is

(A)

isosceles and right angled

(B)

isosceles but not right angled

(C)

right angled but not isosceles

(D)

neither right angled nor isosceles

Answer:A

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let R={(3,3),(6,6),(9,9),(12,12),(6,12),(3,9),(3,12),(3,6)} be a relation on the set A={3,6,9,12}. The relation is

Visualized Solution (English)

Defining Set .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A and Relation .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } R

Checking Reflexivity

Checking Symmetry

Checking Transitivity

Final Conclusion

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