Let R = (1,3), (4,2), (2,4), (2,3),...

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

Visualized Solution (English)

Visualizing the Relation .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } R

Given Set $A = {1, 2, 3, 4}$
Relation $R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)}$

Checking Reflexivity

A relation is reflexive if $(a, a) \in R$ for all $a \in A$ .
Here, $(1, 1) \in / R$ , $(2, 2) \in / R$ , etc.
Therefore, $R$ is not reflexive.

Checking Symmetry

A relation is symmetric if $(a, b) \in R ⟹ (b, a) \in R$ .
Check pairs:
$(1, 3) \in R$ and $(3, 1) \in R$ (Symmetric for this pair)
$(4, 2) \in R$ and $(2, 4) \in R$ (Symmetric for this pair)
But, $(2, 3) \in R$ while $(3, 2) \in / R$ .
Conclusion: $R$ is not symmetric.

Checking Transitivity

A relation is transitive if $(a, b) \in R$ and $(b, c) \in R ⟹ (a, c) \in R$ .
Consider $(1, 3) \in R$ and $(3, 1) \in R$ .
For transitivity, $(1, 1)$ must be in $R$ .
Since $(1, 1) \in / R$ , the relation is not transitive.

Checking Function Property

A relation is a function if every element in the domain has exactly one image.
In $R$ , element $2$ is related to both $3$ and $4$ .
$(2, 3) \in R$ and $(2, 4) \in R$ .
Since one input has multiple outputs, $R$ is not a function.

Conclusion & Final Answer

Key Takeaway: To check relation types, test each definition against the given set of pairs.
Final Result: The relation is not symmetric because $(2, 3) \in R$ but $(3, 2) \in / R$ .
Next Challenge: What minimum pairs must be added to $R$ to make it an equivalence relation?

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Conceptually Similar Problems

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

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

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 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 2002Easy

View in:English Hindi

Let function $f : R \to R$ be defined by $f (x) = 2 x + sin x$ for $x \in R$ , then $f$ is

(A)

one-to-one and onto

(B)

one-to-one but NOT onto

(C)

onto but NOT one-to-one

(D)

neither one-to-one nor onto

Answer:A

MathematicsClassification of FunctionsJEE Main 2009Easy

View in:English Hindi

For real $x$ , let $f (x) = x^{3} + 5 x + 1$ , then

(A)

f

is onto

R

but not one-one

(B)

f

is one-one and onto

R

(C)

f

is neither one-one nor onto

R

(D)

f

is one-one but not onto

R

Answer:B

MathematicsClassification of FunctionsJEE Main 2003Easy

View in:English Hindi

A function $f$ from the set of natural numbers to integers defined by $f (n) = {\frac{n - 1}{2}, - \frac{n}{2}, when n is odd when n is even$ is

(A)

neither one-one nor onto

(B)

one-one but not onto

(C)

onto but not one-one

(D)

one-one and onto

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

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 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

Visualized Solution (English)

Visualizing the 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

Checking Function Property

Conclusion & Final Answer

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