## Chapter 1 - Relations and Functions

#### Pages 5 - 7

Determine whether the following relations are reflective, symmetric and transitive:

Relation R in the set A = {1, 2, 3...13, 14} defined as R = {(x,y):3x - y = 0}

Determine whether each of the following relations are reflexive, symmetric and transitive:

Relation R in the set **N** of natural numbers defined as

R = {(*x*, *y*): *y* = *x* + 5 and *x* < 4}

Determine whether each of the following relations are reflexive, symmetric and transitive:

Relation R in the set *A* = {1, 2, 3, 4, 5, 6} as R = {(*x*, *y*): *y* is divisible by *x*}

Determine whether each of the following relations are reflexive, symmetric and transitive:

Relation R in the set **Z** of all integers defined as

R = {(*x*, *y*): *x* − *y* is as integer}

Relation R in the set *A* of human beings in a town at a particular time given by

(a) R = {(*x*, *y*): *x *and *y* work at the same place}

(b) R = {(*x*, *y*): *x* and *y* live in the same locality}

(c) R = {(*x*, *y*): *x *is exactly 7 cm taller than *y*}

(d) R = {(*x*, *y*): *x* is wife of *y*}

(e) R = {(*x*, *y*): *x* is father of* y*}

Show that the relation R in the set **R **of real numbers, defined as

R = {(*a*, *b*): *a* ≤ *b*^{2}} is neither reflexive nor symmetric nor transitive.

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(*a*, *b*): *b* = *a* + 1} is reflexive, symmetric or transitive.

Show that the relation R in **R** defined as R = {(*a*, *b*): *a* ≤ *b*}, is reflexive and transitive but not symmetric.

Check whether the relation R in **R** defined as R = {(*a*, *b*): *a* ≤ *b*^{3}} is reflexive, symmetric or transitive.

Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

Show that the relation R in the set *A* of all the books in a library of a college, given by R = {(*x*, *y*): *x* and *y* have same number of pages} is an equivalence relation.

Show that the relation R in the set *A* = {1, 2, 3, 4, 5} given by `R = {(a,b) : |a-b| " is even"}` is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.

Show that each of the relation R in the set A= {x in Z : 0 <= x <= 12} given by R = {(a,b):|a-b| is a multiple of 4}

is an equivalence relation. Find the set of all elements related to 1 in each case.

Show that each of the relation R in the set `A = {x =Z: 0 <= <= 12 }` given by `R = {(a,b) : a= b}

is an equivalence relation. Find the set of all elements related to 1 in each case.

Given an example of a relation. Which is Symmetric but neither reflexive nor transitive.

Given an example of a relation. Which is Transitive but neither reflexive nor symmetric.

Given an example of a relation. Which is Reflexive and symmetric but not transitive.

Given an example of a relation. Which is Reflexive and transitive but not symmetric.

Given an example of a relation. Which is Symmetric and transitive but not reflexive.

Show that the relation R in the set *A* of points in a plane given by R = {(P, Q): distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all point related to a point P ≠ (0, 0) is the circle passing through P with origin as centre.

Show that the relation R defined in the set *A* of all triangles as R = {(*T*_{1}, *T*_{2}): *T*_{1} is similar to *T*_{2}}, is equivalence relation. Consider three right angle triangles *T*_{1} with sides 3, 4, 5, *T*_{2} with sides 5, 12, 13 and *T*_{3} with sides 6, 8, 10. Which triangles among *T*_{1}, *T*_{2}and *T*_{3} are related?

Show that the relation R defined in the set *A* of all polygons as R = {(*P*_{1}, *P*_{2}): *P*_{1} and *P*_{2}have same number of sides}, is an equivalence relation. What is the set of all elements in *A* related to the right angle triangle *T* with sides 3, 4 and 5?

Let *L* be the set of all lines in XY plane and R be the relation in *L* defined as R = {(*L*_{1}, *L*_{2}): *L*_{1} is parallel to *L*_{2}}. Show that R is an equivalence relation. Find the set of all lines related to the line *y* = 2*x* + 4.

Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer.

(A) R is reflexive and symmetric but not transitive.

(B) R is reflexive and transitive but not symmetric.

(C) R is symmetric and transitive but not reflexive.

(D) R is an equivalence relation.

Let R be the relation in the set **N **given by R = {(*a*, *b*): *a *= *b* − 2, *b *> 6}. Choose the correct answer.

(A) (2, 4) ∈ R

(B) (3, 8) ∈ R

(C) (6, 8) ∈ R

(D) (8, 7) ∈ R

#### Pages 10 - 11

Show that the function *f*: **R**_{*} → **R**_{*} defined by `f(x) = 1/x` is one-one and onto, where **R**_{*} is the set of all non-zero real numbers. Is the result true, if the domain **R**_{*} is replaced by **N** with co-domain being same as **R**?

Check the injectivity and surjectivity of the following functions:

*f*: **N** → **N** given by *f*(*x*) = *x*^{2}

Check the injectivity and surjectivity of the given functions: *f*: **Z** → **Z** given by *f*(*x*) = *x*^{2}

Check the injectivity and surjectivity of the given functions: *f*: **R** → **R** given by *f*(*x*) = *x*^{2}

Check the injectivity and surjectivity of the following functions: *f*: **N **→ **N** given by *f*(*x*) = *x*^{3}

Check the injectivity and surjectivity of the following functions: *f*: **Z** → **Z** given by *f*(*x*) = *x*^{3}

Prove that the Greatest Integer Function *f*: **R **→ **R** given by *f*(*x*) = [*x*], is neither one-once nor onto, where [*x*] denotes the greatest integer less than or equal to *x*.

Show that the Modulus Function *f*: **R **→ **R** given by `f(x) = |x|` is neither one-one nor onto, where `|x|` is *x*, if *x*is positive or 0 and |x| is − *x*, if *x* is negative.

Show that the Signum Function* f*: **R** → **R**, given by

Let *A* = {1, 2, 3}, *B *= {4, 5, 6, 7} and let *f* = {(1, 4), (2, 5), (3, 6)} be a function from *A* to *B*. Show that *f* is one-one.

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

*f*: **R **→ **R** defined by *f*(*x*) = 3 − 4*x*

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

*f*: **R **→ **R** defined by *f*(*x*) = 1 + *x*^{2}

Let *A* and *B* be sets. Show that *f*: *A* × *B* → *B *× *A* such that (*a*, *b*) = (*b*, *a*) is bijective function.

Let *f*: **N** → **N** be defined by f(n) = `{((n+1)/2, "if n is odd"),(," for all n ∈ N"), (n/2, if "n is even"):}`

State whether the function f is bijective. Justify your answer.

Let A = **R** − {3} and B = **R** − {1}. Consider the function *f*: A → B defined by `f(x) = ((x- 2)/(x -3))`. Is *f* one-one and onto? Justify your answer.

Let *f*: **R** → **R** be defined as* f*(*x*) = *x*^{4}. Choose the correct answer.

(A) *f* is one-one onto

(B) *f* is many-one onto

(C) *f* is one-one but not onto

(D) *f* is neither one-one nor onto

Let *f*: **R** → **R** be defined as *f*(*x*) = 3*x*. Choose the correct answer.

(A) *f* is one-one onto

(B) *f* is many-one onto

(C) *f* is one-one but not onto

(D) *f* is neither one-one nor onto

#### Pages 18 - 19

Let *f*: {1, 3, 4} → {1, 2, 5} and *g*: {1, 2, 5} → {1, 3} be given by *f *= {(1, 2), (3, 5), (4, 1)} and *g *= {(1, 3), (2, 3), (5, 1)}. Write down *g*o*f*.

Let *f*, *g* and *h* be functions from **R **to **R**. Show that

`(f + g)oh = foh + goh`

`(f.g)oh = (foh).(goh)`

Find gof and fog, if f(x) = |x| and g(x) = |5x - 2|

Find *g*o*f *and *f*o*g*, if `f(x) = 8x^3` and `g(x) = x^(1/3)`

if f(x) = `(4x + 3)/(6x - 4), x != 2/3` show that* f*o*f*(*x*) = *x*, for all x != 2/3 . What is the inverse of *f*?

State with reason whether following functions have inverse

*f*: {1, 2, 3, 4} → {10} with

*f* = {(1, 10), (2, 10), (3, 10), (4, 10)}

State with reason whether following functions have inverse *g*: {5, 6, 7, 8} → {1, 2, 3, 4} with *g* = {(5, 4), (6, 3), (7, 4), (8, 2)}

State with reason whether following functions have inverse *h*: {2, 3, 4, 5} → {7, 9, 11, 13} with *h* = {(2, 7), (3, 9), (4, 11), (5, 13)}

Show that *f*: [−1, 1] → **R**, given by f(x) = `x/(x + 2)` is one-one. Find the inverse of the function *f*: [−1, 1] → Range *f*.

(Hint: For y in Range f, y = `f(x) = x/(x +2)` for some x in [-1, 1] ie x = `2y/(1-y)`

Consider *f*: **R** → **R** given by *f*(*x*) = 4*x* + 3. Show that *f* is invertible. Find the inverse of *f*.

Consider *f*: **R**_{+ }→ [4, ∞) given by *f*(*x*) =* x*^{2} + 4. Show that *f* is invertible with the inverse *f*^{−1} of given *f *by `f^(-1) (y) = sqrt(y - 4)` where **R**_{+} is the set of all non-negative real numbers.

Consider *f*: **R**_{+} → [−5, ∞) given by *f*(*x*) = 9*x*^{2} + 6*x* − 5. Show that *f* is invertible with `f^(-1)(y) = ((sqrt(y +6) - 1)/3)`

Let *f*: *X* → *Y* be an invertible function. Show that* f *has unique inverse. (Hint: suppose *g*_{1} and *g*_{2} are two inverses of *f*. Then for all *y* ∈ *Y*, *f*o*g*_{1}(*y*) = I_{Y}(*y*) = *f*o*g*_{2}(*y*). Use one-one ness of *f*).

Consider *f*: {1, 2, 3} → {*a*, *b*, *c*} given by *f*(1) = *a*, *f*(2) = *b* and *f*(3) = *c*. Find *f*^{−1} and show that (*f*^{−1})^{−1} =* f*.

Let *f*: *X* → *Y* be an invertible function. Show that the inverse of *f*^{−1} is *f*, i.e., (*f*^{−1})^{−1} = *f*.

If* f*: **R **→ **R **be given by `f(x) = (3 - x^3)^(1/3)` , then *f*o*f*(*x*) is

(A) `1/(x^3)`

(B) *x*^{3}

(C) *x*

(D) (3 − *x*^{3})

Let `f:R - {-4/3} -> R` be a function defined as `f(x) = (4x)/(3x + 4)`. The inverse of *f* is map *g* Range `f -> R -{- 4/3}`

(A) `g(y) = (3y)/(3-4y)`

(B) `g(y) = (4y)/(4 - 3y)`

(C) `g(y) = (4y)/(3 - 4y)`

(D) `g(y) = (3y)/(4 - 3y)`

#### Pages 24 - 26

Determine whether or not of the definition of ∗ given below gives a binary operation. In the event that ∗ is not a binary operation, give justification for this.

On Z+, define ∗ by a ∗ b = a – b

Determine whether or not of the definition of given below gives a binary operation. In the event that * is not a binary operation, give justification for this.

On **Z**^{+}, define * by *a ** *b *= *ab*

Determine whether or not each of the definition of given below gives a binary operation. In the event that * is not a binary operation, give justification for this.

On **R**, define * by *a ** *b *= *ab*^{2}

Determine whether or not each of the definition of given below gives a binary operation. In the event that * is not a binary operation, give justification for this.

On **Z**^{+}, define * by *a ** *b *= |*a *− *b*|

Determine whether or not each of the definition of given below gives a binary operation. In the event that * is not a binary operation, give justification for this.

On **Z**^{+}, define * by *a ** *b *= *a*

For each binary operation * defined below, determine whether * is commutative or associative.

On **Z**, define *a ** *b *= *a *− *b*

For each binary operation * defined below, determine whether * is commutative or associative.

On **Q**, define *a ** *b *= *ab *+ 1

For each binary operation * defined below, determine whether * is commutative or associative.

On **Q**, define a * b = `(ab)/2`

For each binary operation * defined below, determine whether * is commutative or associative.

On **Z**^{+}, define *a ** *b *= 2^{ab}

For each binary operation * defined below, determine whether * is commutative or associative.

On **Z**^{+}, define *a ** *b *= *a*^{b}

For each binary operation * defined below, determine whether * is commutative or associative.

On **R **− {−1}, define `a*b = a/(b+1)`

Consider the binary operation ∨ on the set {1, 2, 3, 4, 5} defined by *a *∨*b *= min {*a*, *b*}. Write the operation table of the operation∨.

Consider a binary operation * on the set {1, 2, 3, 4, 5} given by the following multiplication table.

(i) Compute (2 * 3) * 4 and 2 * (3 * 4)

(ii) Is * commutative?

(iii) Compute (2 * 3) * (4 * 5).

(Hint: use the following table)

* | 1 | 2 | 3 | 4 | 5 |

1 | 1 | 1 | 1 | 1 | 1 |

2 | 1 | 2 | 1 | 2 | 1 |

3 | 1 | 1 | 3 | 1 | 1 |

4 | 1 | 2 | 1 | 4 | 1 |

5 | 1 | 1 | 1 | 1 | 5 |

Let*′ be the binary operation on the set {1, 2, 3, 4, 5} defined by *a **′ *b *= H.C.F. of *a *and *b*. Is the operation *′ same as the operation * defined in Exercise 4 above? Justify your answer.

Let * be the binary operation on **N **given by *a* * *b *= L.C.M. of *a *and *b*. Find

(i) 5 * 7, 20 * 16

(ii) Is * commutative?

(iii) Is * associative?

(iv) Find the identity of * in **N**

(v) Which elements of **N **are invertible for the operation *?

Is * defined on the set {1, 2, 3, 4, 5} by *a ** *b *= L.C.M. of *a *and *b *a binary operation? Justify your answer.

Let * be a binary operation on the set **Q **of rational numbers as follows:

(i) *a ** *b *= *a *− *b *

(ii) *a ** *b *= *a*^{2} + *b*^{2}

(iii) *a ** *b *= *a *+ *ab *

(iv) *a ** *b *= (*a *− *b*)^{2}

(v) a * b = ab/4

(vi) *a ** *b *= *ab*^{2}

Find which of the binary operations are commutative and which are associative.

Find which of the operations given above has identity.

Let A = **N **× **N **and * be the binary operation on A defined by (*a*, *b*) * (*c*, *d*) = (*a *+ *c*, *b *+ *d*)

Show that * is commutative and associative. Find the identity element for * on A, if any.

State whether the following statements are true or false. Justify.

For an arbitrary binary operation * on a set **N**, *a ** *a *= ∀ *a * *a ** **N**.

State whether the following statements are true or false. Justify.

If * is a commutative binary operation on **N**, then *a ** (*b ** *c*) = (*c ** *b*) * *a*

Consider a binary operation * on **N **defined as *a ** *b *= *a*^{3} + *b*^{3}. Choose the correct answer.

(A) Is * both associative and commutative?

(B) Is * commutative but not associative?

(C) Is * associative but not commutative?

(D) Is * neither commutative nor associative?

#### Pages 29 - 31

Let *f*: **R** → **R **be defined as *f*(*x*) = 10*x* + 7. Find the function *g*: **R** → **R** such that *g* o *f* = *f* o *g *= 1_{R}.

Let *f*: W → W be defined as *f*(*n*) = *n* − 1, if is odd and *f*(*n*) = *n* + 1, if *n* is even. Show that* f* is invertible. Find the inverse of *f*. Here, W is the set of all whole numbers.

If *f*: **R **→ **R** is defined by *f*(*x*) = *x*^{2} − 3*x *+ 2, find *f*(*f*(*x*)).

Show that function *f*: **R** → {*x* ∈ **R**: −1 < *x* < 1} defined by *f*(*x*) =`x/(1 + |x|)` , *x *∈**R** is one-one and onto function.

Show that the function *f*: **R** → **R** given by *f*(*x*) = *x*^{3} is injective.

Give examples of two functions *f*: **N** → **Z** and *g*: **Z** → **Z** such that *g* o *f* is injective but *g*is not injective.

(Hint: Consider *f*(*x*) = *x* and *g*(*x*) =|x|)

Given examples of two functions *f*: **N** → **N** and *g*: **N** → **N** such that *g*o*f* is onto but *f *is not onto.

(Hint: Consider *f*(*x*) = *x* + 1 and `g(x) = {(x-1, ifx >1),(1, if x = 1):}`

Given a non empty set *X*, consider P(*X*) which is the set of all subsets of *X*.

Define the relation R in P(*X*) as follows:

For subsets *A*, *B* in P(*X*), *A*R*B* if and only if *A* ⊂ *B*. Is R an equivalence relation on P(*X*)? Justify you answer:

Given a non-empty set *X*, consider the binary operation *: P(*X*) × P(*X*) → P(*X*) given by *A* * *B* = *A* ∩ *B* &mnForE; *A*, *B* in P(*X*) is the power set of *X*. Show that *X *is the identity element for this operation and *X *is the only invertible element in P(*X*) with respect to the operation*.

Find the number of all onto functions from the set {1, 2, 3, … , *n*) to itself.

Let *S* = {*a*, *b*, *c*} and *T* = {1, 2, 3}. Find F^{−1} of the following functions F from *S* to *T*, if it exists.

*F* = {(*a*, 3), (*b*, 2), (*c*, 1)}

Let *S* = {*a*, *b*, *c*} and *T* = {1, 2, 3}. Find F^{−1} of the following functions F from *S* to *T*, if it exists.

*F* = {(*a*, 2), (*b*, 1), (*c*, 1)}

Consider the binary operations*: **R **×**R **→ and o: **R** × **R** → **R **defined as a * b = |a - b| and *a*o *b* = *a*, &mnForE;*a*, *b* ∈ **R**. Show that * is commutative but not associative, o is associative but not commutative. Further, show that &mnForE;*a*, *b*, *c* ∈ **R**, *a**(*b* o *c*) = (*a** *b*) o (*a* * *c*). [If it is so, we say that the operation * distributes over the operation o]. Does o distribute over *? Justify your answer.

Given a non-empty set *X*, let *: P(*X*) × P(*X*) → P(*X*) be defined as *A* * *B* = (*A* − *B*) ∪ (*B* −*A*), &mnForE; *A*, *B* ∈ P(*X*). Show that the empty set *Φ* is the identity for the operation * and all the elements *A* of P(*X*) are invertible with *A*^{−1} = *A*. (Hint: (*A* − *Φ*) ∪ (*Φ* − *A*) = *A*and (*A* − *A*) ∪ (*A* − *A*) = *A* * *A* = *Φ*).

Define a binary operation *on the set {0, 1, 2, 3, 4, 5} as

a * b = `{(a+b, "if a+b < 6"), (a + b - 6, if a +b >= 6):}`

Show that zero is the identity for this operation and each element *a* ≠ 0 of the set is invertible with 6 − *a* being the inverse of *a*.

Let *A* = {−1, 0, 1, 2}, *B* = {−4, −2, 0, 2} and *f*, *g*: *A* → *B* be functions defined by *f*(*x*) = *x*^{2} − *x*, *x* ∈ A and g(x) = `2|x - 1/2|- 1, x in A`. Are *f* and *g* equal?

Justify your answer. (Hint: One may note that two function *f*: *A* → *B* and g: *A* → B such that *f*(*a*) = g(*a*) &mn For E;*a* ∈*A*, are called equal functions).

Let *A* = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is

(A) 1

(B) 2

(C) 3

(D) 4

Let *A* = {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is

(A) 1 (B) 2 (C) 3 (D) 4

Let *f*: **R** → **R** be the Signum Function defined as

f(x) = `{(1,x>0), (0, x =0),(-1, x< 0):}`

and *g*: **R** → **R **be the Greatest Integer Function given by *g*(*x*) = [*x*], where [*x*] is greatest integer less than or equal to *x*. Then does *f*o*g* and *g*o*f* coincide in (0, 1]?

Number of binary operations on the set {*a*, *b*} are

(A) 10

(B) 16

(C) 20

(D) 8