Question

2. Represent all possible one-one functions from the set A = {1, 2} to
the set B = {3,4,5) using arrow diagram.
(3)​

Answer 1

Given :   set A = {1, 2} to  the set B = {3,4,5)  

To Find : Represent all possible one-one functions from the set A  to Set B

Solution:

A = { 1 , 2}

B = { 3 , 4 , 5 }

n(A) = 2

n(B) = 3

Number pf one to one functions =  ³P₂  = 6

(1 →3) , ( 2→ 4)  

(1 → 3) , ( 2→ 5)

( 1→ 4) , ( 2→3)

( 1→ 4) , ( 2→ 5)

(1→ 5)  , ( 2→ 3)

(1→ 5) , ( 2→4)

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Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

DEFINITION TO BE MEMORISED

RELATION

Let A & B be any two non empty sets. A relation R between A & B is a subset of A × B

MAPPING

Let A & B be any two non empty sets. A relation f with the property that each element x of A is related to exactly one element y of B is said to be a mapping from A to B

ONE - ONE FUNCTION

$\sf{A \: function \: f : A \to \: B \: \: is \: said \: to \: be \: one \: to \: one \: if \: }$

for each pair of distinct elements of A their f - images are distinct

NUMBER OF ONE - ONE FUNCTIONS

$\sf{If \: n(A) = p \: and \: n(B) = q \: with \: p \leqslant q }$

Then total number of one-one functions from A to B

$\displaystyle \: = \sf{\frac{q \: !}{(q - p)!} }$

TO DETERMINE

Represent all possible one-one functions from the set A = {1, 2} to the set B = {3,4,5) using arrow diagram

CALCULATION

Here A × B = {(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)}

$\sf{n(A) = 2 \: and \: n(B) = 3 \: }$

So total number of one-one functions from A to B is

$\displaystyle \: = \sf{\frac{3 \: !}{(3 - 2)!} } = 6$

So the required one - one functions are

$1. \: \: \sf{ \: 1 \to \: 3 \: \:, \: \: 2 \to \: 4\: }$

$2. \: \: \sf{ \: 1 \to \: 3 \: \:, \: \: 2 \to \: 5\: }$

$3. \: \: \sf{ \: 1 \to \: 4 \: \:, \: \: 2 \to \: 3\: }$

$4. \: \: \sf{ \: 1 \to \: 4 \: \:, \: \: 2 \to \: 5\: }$

$5. \: \: \sf{ \: 1 \to \: 5 \: \:, \: \: 2 \to \: 3\: }$

$6. \: \: \sf{ \: 1 \to \: 5 \: \:, \: \: 2 \to \: 4\: }$

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