Relation and functions L1: 12th Elite -DPP

Q1. A relation on A = {-3, -1, 0, 1, 3} is defined as R = {(x, y) : y = |x|, x ≠ -1}Then find number of elements in power set of R.

Q2. Let A = {x, y, z} and B = {a, b}. Find the total number of relations from A into B.

Q3. If the relation R : A → B, where A = {1, 2, 3, 4} and B = {1, 3, 5} is defined by R = {(x, y); x < y, x ∈ A, y ∈ B} then RoR-1 is

A

B

D

C

A

B

D

C

D

{(1, 3), (1, 5), (2, 3), (2, 5), (3, 5), (4, 5)}

{(3, 3), (3, 5), (5, 3), (5, 5)}

{(3, 1), (5, 1), (5, 2), (5, 3), (5, 4)}

None of these

Q4. If R is the relation “less than” from A = {1, 2, 3, 4, 5} to B = {1, 4, 5}, write down the set of ordered pairs corresponding to R. Find the inverse of R.

Q5. Let A be the set of first ten natural numbers and let R be a relation on A defined by (x, y) ∈ R ⇔ x + 2y = 10. Express R and R-1 as sets of ordered pairs.

Q6. Prove that relation R = { (x, y) : x is factor of y } is a Transitive relation on integers.

Q7. Determine whether following relation is reflexive, symmetric and transitive: Relation R in the set A = {1, 2, 3, ……….., 13, 14} defined as R = {(x, y): 3x - y =0}

Reflexive and transitive

Reflexive and symmetric

Symmetric and transitive

Equivalence relation.

A

B

D

C

A

B

D

C

D

Q8. Prove that relation R = { (x, y) : x is factor of y } is a Transitive relation on integers.

A

B

D

C

A

B

D

C

D

Reflexive and transitive only

A equivalence relation

Reflexive only

Reflexive and symmetric only

Q9. Let R = {(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)} be a relation on A = {3, 6, 9, 12}. Relation R is _____.

A

B

D

C

A

B

D

C

D

Neither R nor S is an equivalence relation

R and S both are equivalence relations

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

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

AIEEE 2010

Q10. Consider the following relations: R = {(x, y) | x, y are real numbers and x = wy for some rational number w};

Q11. Let R be the real line. Consider the following subsets of the plane R × R;S = {(x, y): y = x + 1 and 0 < x < 2}T = {(x, y): x - y is an integer} Statement-1: T is an equivalence relation on R but S is not an equivalence relation on R. Statement-2: S is neither reflexive nor symmetric but T is reflexive, symmetric and transitive.

Q12. Check injectivity: y = 2x3 + 3x2 + 6x + 5

Q13. Check whether the following functions are onto or into

defined as

A

B

D

C

A

B

D

C

D

f : R → R, f(x) = 2x + 5

f : [0, π] → [-1, 1], f(x) = cosx

f : R → [-1, 1], f(x) = sinx

Q14. Which of the following functions is not one-one.

Q15. Check whether the following function is onto or into

Q16. Check mapping for the following function

A

B

D

C

A

B

D

C

D

Onto and one-one

Onto and many-one

Into and one-one

Into and many-one

Q17. The function f: R → R given by f(x) = 3 -2 sinx is

A

B

D

C

A

B

D

C

D

One-one

Bijective

Onto

None of these

Q18. The function f: N → N (N is the set of natural numbers), defined by f(n) = 2n + 3 is

A

B

D

C

A

B

D

C

D

Surjective only

Bijective

Injective only

None of these

Q19. Let R be the set of real numbers. If f : R → R is a function defined by f(x) = x2, then f is

A

B

D

C

A

B

D

C

D

Injective but not Surjective

Bijective

Surjective but not injective

None of these

IIT-JEE, 1979

Q20. Which of the following functions is not onto

A

B

D

C

A

B

D

C

D

f : R → R, f(x) = 3x + 4

f : R → R+, f(x) = x2 + 2

None of these

Q21. Let X = Y = R ~ {1}. The function f : X → Y defined by is

A

B

D

C

A

B

D

C

D

One-one but not onto

Neither one-one nor onto

Onto but not one-one

One-one and onto

Q22. Let f : (1, ∞) → (1, ∞) be defined by

A

B

D

C

A

B

D

C

D

f is 1 -1 and onto

f is not 1 - 1 but onto

f is 1 -1 but not onto

f is neither 1 - 1 nor onto

AIEEE 2008

Q23. Let f : R → R be a function defined by f(x) = x2009 + 2009x + 2009 . Then f(x) is

A

B

D

C

A

B

D

C

D

One-one but not onto

Neither one-one nor onto

Not one-one but onto

One-one and onto

AIEEE 2010

Q24. Show that f : N ➝ N, defined by is a many-one onto function.

Relation and functions L1: 12th Elite - DPP Solutions

Q1. A relation on A = {-3, -1, 0, 1, 3} is defined as R = {(x, y) : y = |x|, x ≠ -1}Then find number of elements in power set of R.

R = {(-3, 3), (0, 0), (1, 1), (3, 3)}

Number of elements in power set of R = 24 = 32.

Solution:

Q2. Let A = {x, y, z} and B = {a, b}. Find the total number of relations from A into B.

Solution:

Q3. If the relation R : A → B, where A = {1, 2, 3, 4} and B = {1, 3, 5} is defined by R = {(x, y); x < y, x ∈ A, y ∈ B} then RoR-1 is

A

B

D

C

A

B

D

C

D

{(1, 3), (1, 5), (2, 3), (2, 5), (3, 5), (4, 5)}

{(3, 3), (3, 5), (5, 3), (5, 5)}

{(3, 1), (5, 1), (5, 2), (5, 3), (5, 4)}

None of these

Solution:

Q4. If R is the relation “less than” from A = {1, 2, 3, 4, 5} to B = {1, 4, 5}, write down the set of ordered pairs corresponding to R. Find the inverse of R.

Solution:

Q5. Let A be the set of first ten natural numbers and let R be a relation on A defined by (x, y) ∈ R ⇔ x + 2y = 10. Express R and R-1 as sets of ordered pairs.

Solution:

Solution:

Q6. Prove that relation R = { (x, y) : x is factor of y } is a Transitive relation on integers.

Let (a, b) ∈ R and (b, c) ∈ R

∴ b = ak1 and c = bk

2 for some k

1, k

2 ∈ Z

⇒ c = ak1k

2

∴ a is a factor of c.

∴ (a, c) ∈ R .

Hence it is transitive relation

Solution:

Q7. Determine whether following relation is reflexive, symmetric and transitive: Relation R in the set A = {1, 2, 3, ……….., 13, 14} defined as R = {(x, y): 3x - y =0}

R = {(x, y) : 3x - y = 0}

A = {1, 2, 3, 4, 5, 6, …… 13, 14}

Therefore, R = {(1, 3), (2, 6), (3, 9), (4, 12)} … (1)

As per reflexive property : (x, x) ∈ R, then R is reflexive

Since there is no such pair, R is not reflexive.

As per symmetric property : (x, y) ∈ R and (y, x) ∈ R, then R is symmetric.

Since there is no such pair, R is not symmetric

As per transitive property : If (x, y) ∈ R and (y, z) ∈ R, then (x, z) ∈ R. Thus R

is transitive.

Solution:

From (1), (1, 3) ∈ R and (3, 9) ∈ R but (1, 9) ∉ R, R is not transitive.

Therefore, R is neither reflexive, nor symmetric and nor transitive.

Solution:

Reflexive and transitive

Reflexive and symmetric

Symmetric and transitive

Equivalence relation.

Q8. Prove that relation R = { (x, y) : x is factor of y } is a Transitive relation on integers.

A

B

D

C

A

B

D

C

D

Solution:

Q9. Let R = {(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)} be a relation on A = {3, 6, 9, 12}. Relation R is _____.

A

B

D

C

A

B

D

C

D

Reflexive and transitive only

A equivalence relation

Reflexive only

Reflexive and symmetric only

Given A = {3, 6, 9, 12}

Since (3, 3), (6, 6), (9, 9), (12, 12) ∈ R

Thus its reflexive relation

Here, (6, 12) ∈ R, but (12, 6) ∉ R

Thus R is not symmetric

Here, (3, 6) ∈ R and (6, 12) ∈ R, also (3, 12) ∈ R

So relation is transitive also.

Solution:

Q10. Consider the following relations: R = {(x, y) | x, y are real numbers and x = wy for some rational number w};

A

B

D

C

A

B

D

C

D

Neither R nor S is an equivalence relation

R and S both are equivalence relations

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

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

AIEEE 2010

Solution:

Q11. Let R be the real line. Consider the following subsets of the plane R × R;S = {(x, y): y = x + 1 and 0 < x < 2}T = {(x, y): x - y is an integer} Statement-1: T is an equivalence relation on R but S is not an equivalence relation on R. Statement-2: S is neither reflexive nor symmetric but T is reflexive, symmetric and transitive.

Solution:

Q12. Check injectivity: y = 2x3 + 3x2 + 6x + 5

Solution:

Q13. Check whether the following functions are onto or into

defined as

Taking D ≥ 0

Thus f(x) is onto.

Solution:

A

B

D

C

A

B

D

C

D

f : R → R, f(x) = 2x + 5

f : [0, π] → [-1, 1], f(x) = cosx

f : R → [-1, 1], f(x) = sinx

Q14. Which of the following functions is not one-one.

Solution:

Q15. Check whether the following function is onto or into

Here,

Thus f(x) is onto.

∴

Solution:

Q16. Check mapping for the following function

A

B

D

C

A

B

D

C

D

Onto and one-one

Onto and many-one

Into and one-one

Into and many-one

Since

Range of

Thus it is onto function

Also f(x) gives same value for

Thus f(x) is many-one

Solution:

Q17. The function f: R → R given by f(x) = 3 -2 sinx is

A

B

D

C

A

B

D

C

D

One-one

Bijective

Onto

None of these

Solution:

Q18. The function f: N → N (N is the set of natural numbers), defined by f(n) = 2n + 3 is

A

B

D

C

A

B

D

C

D

Surjective only

Bijective

Injective only

None of these

Solution:

Q19. Let R be the set of real numbers. If f : R → R is a function defined by f(x) = x2, then f is

A

B

D

C

A

B

D

C

D

Injective but not Surjective

Bijective

Surjective but not injective

None of these

IIT-JEE, 1979

Solution:

f(x) = x2 is many-one as f(1) = f(-1) = 1.

Also f is into, as the range of function is [0, ∞)

which is subset of R (co-domain).

∴ f is neither injective nor surjective.

Q20. Which of the following functions is not onto

A

B

D

C

A

B

D

C

D

f : R → R, f(x) = 3x + 4

f : R → R+, f(x) = x2 + 2

None of these

Solution:

Q21. Let X = Y = R ~ {1}. The function f : X → Y defined by is

A

B

D

C

A

B

D

C

D

One-one but not onto

Neither one-one nor onto

Onto but not one-one

One-one and onto

Solution:

Q22. Let f : (1, ∞) → (1, ∞) be defined by

A

B

D

C

A

B

D

C

D

f is 1 -1 and onto

f is not 1 - 1 but onto

f is 1 -1 but not onto

f is neither 1 - 1 nor onto

AIEEE 2008

Solution:

Q23. Let f : R → R be a function defined by f(x) = x2009 + 2009x + 2009 . Then f(x) is

A

B

D

C

A

B

D

C

D

One-one but not onto

Neither one-one nor onto

Not one-one but onto

One-one and onto

AIEEE 2010

Solution:

f is continuous function and f’(x) = 2009 x2008 + 2009 > 0.

So f is one-one and onto.

Q24. Show that f : N ➝ N, defined by is a many-one onto function.

Solution:

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