Chapter 1 : Relations and Functions

Let

f, g

and

h

be functions from

R

R

. Show that

(i)

(f + g) o h = f o h + g o h

(ii)

(f . g) o h = (f o h) . (g o h)

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Q. Symmetric and transitive but not reflexive.

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

(i) $f : R \to R$ defined by $f (x) = 3 - 4 x$

(ii) $f : R \to R$ defined by $f (x) = 1 + x^{2}$

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Show that each of the relation

R

in the set

A = {x \in Z : 0 \leq x \leq 12}

, given by

(i)

R = {(a, b) : | a - b | is a multiple of 4}

(ii)

R = {(a, b) : a = b}

is an equivalence relation. Find the set of all elements related to

1

in each case.

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Determine whether the following relation is reflexive, symmetric and transitive

Relative

R

in the set

N

of natural numbers defined as

R = {(x, y) : y = x + 5 a n d x < 4}

enter 1-Reflexive
2-Symmetric
3-Transitive
4-Equivalence
5-None

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Q. Relation

R

in the set

A

of human beings in a town at a particular time given by

R = {(x, y) : x i s f a t h e r o f y}

enter 1-reflexive and transitive but not symmetric

2-reflexive only

3-Transitive only

4-Equivalence

5-Neither reflexive, nor symmetric, nor transitive

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Q. Relation

R

in the set

Z

of all integers defined as

R = {(x, y) : (x - y) i s a n i n t e g e r}

enter 1-reflexive and transitive but not symmetric

2-reflexive only

3-Transitive only

4-Equivalence

5-None

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

Relation

R

in the set

A = {1, 2, 3, 4, 5, 6}

R = {(x, y) : y is divisible by x}

enter 1-reflexive and transitive but not symmetric

2-reflexive only

3-Transitive only

4-Equivalence

5-None

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

R = {(x, y) : x i s e x a c t l y 7 c m t a l l e r t h a n y}

enter 1-reflexive and transitive but not symmetric

2-reflexive only

3-Transitive only

4-Equivalence

5-Neither reflexive, nor symmetric, nor transitive

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Given an example of a relation. Which is

(i) Symmetric but neither reflexive nor transitive.

(ii) Transitive but neither reflexive nor symmetric.

(iii) Reflexive and symmetric but not transitive.

(iv) Reflexive and transitive but not symmetric.

(v) Symmetric and transitive but not reflexive.

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Q. Check the injectivity and surjectivity of the following functions:

f : Z \to Z

given by

f (x) = x^{2}

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

R = {(x, y) : x a n d y l i v e i n t h e s a m e l o c a l i t y}

enter 1-reflexive and transitive but not symmetric

2-reflexive only

3-Transitive only

4-Equivalence

5-None

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

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Q. Relation

R

in the set

A

of human beings in a town at a particular time given by

R = {(x, y) : x i s w i f e o f y}

enter 1-reflexive and transitive but not symmetric

2-reflexive only

3-Transitive only

4-Equivalence

5-Neither reflexive, nor symmetric, nor transitive

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Show that each of the relation

R

in the set

A = {x \in Z : 0 \leq x \leq 12}

, given by is an equivalence relation. Find the set of all elements related to

1

in each case.

R = {(a, b) : a = b}

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

(i) Relation

R

in the set

A = {1, 2, 3, . . ., 13, 14}

defined as

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

(ii) Relative

R

in the set

N

of natural numbers defined as

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

(iii) Relation

R

in the set

A = {1, 2, 3, 4, 5, 6}

R = {(x, y) : y is divisible by x}

(iv) Relative

R

in the set

Z

of all integers defined as

R = {(x, y) : x - y is an integer}

(v) 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}

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Check the injectivity and surjectivity of the following functions:

(i)

f : N \to N

given by

f (x) = x^{2}

(ii)

f : Z \to Z

given by

f (x) = x^{2}

(iii)

f : R \to R

given by

f (x) = x^{2}

(iv)

f : N \to N

given by

f (x) = x^{3}

(v)

f : Z \to Z

given by

f (x) = x^{3}

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

R = {(x, y) : x a n d y w o r k a t t h e s a m e p l a c e}

enter 1-reflexive and transitive but not symmetric

2-reflexive only

3-Transitive only

4-Equivalence

5-None

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Q. Check whether the relation

R

defined in the set

{1, 2, 3, 4, 5, 6}

R = {(a, b) : b = a + 1}

is reflexive, symmetric or transitive.

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Q. Show that the relation

R

in the set

R

of real numbers, defined as

R = {(a, b) : a \leq b^{2}}

is neither reflexive nor symmetric nor transitive.

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