Question

List all the bijections from A into B.

a) A={q,r,s} and B={2,3,4}

b) A={1,2,3,4} and B={5,6,7,8}

Answer 1

The list of bijections from A into B are shown below.

Explanation:

A function f is called one-to-one or injective, if and only if

`f(x)=f(y)\Rightarrow x = y`

for all x and y in the domain of f.

A function f from X to Y is called onto or surjective, if and only if

for every element y∈Y there is an element x∈X with f(x)=y.

If a function is one-one and onto, then it is called bijective.

Part (a):

A={q,r,s} and B={2,3,4}

We need to find all the bijections from A into B.

(1)
`A\rightarrow B=\{(q,2),(r,3),(s,4)\}`

(2)
`A\rightarrow B=\{(q,2),(r,4),(s,3)\}`

(3)
`A\rightarrow B=\{(q,3),(r,2),(s,4)\}`

(4)
`A\rightarrow B=\{(q,3),(r,4),(s,2)\}`

(5)
`A\rightarrow B=\{(q,4),(r,2),(s,3)\}`

(6)
`A\rightarrow B=\{(q,4),(r,3),(s,2)\}`

Part (b):

A={1,2,3,4} and B={5,6,7,8}

We need to find all the bijections from A into B.

(1)
`A\rightarrow B=\{(1,5),(2,6),(3,7),(4,8)\}`

(2)
`A\rightarrow B=\{(1,5),(2,6),(3,8),(4,7)\}`

(3)
`A\rightarrow B=\{(1,5),(2,7),(3,6),(4,8)\}`

(4)
`A\rightarrow B=\{(1,5),(2,7),(3,8),(4,6)\}`

(5)
`A\rightarrow B=\{(1,5),(2,8),(3,6),(4,7)\}`

(6)
`A\rightarrow B=\{(1,5),(2,8),(3,7),(4,6)\}`

(7)
`A\rightarrow B=\{(1,6),(2,5),(3,7),(4,8)\}`

(8)
`A\rightarrow B=\{(1,6),(2,5),(3,8),(4,7)\}`

(9)
`A\rightarrow B=\{(1,6),(2,7),(3,5),(4,8)\}`

(10)
`A\rightarrow B=\{(1,6),(2,7),(3,8),(4,5)\}`

(11)
`A\rightarrow B=\{(1,6),(2,8),(3,5),(4,7)\}`

(12)
`A\rightarrow B=\{(1,6),(2,8),(3,7),(4,5)\}`

(13)
`A\rightarrow B=\{(1,7),(2,6),(3,5),(4,8)\}`

(14)
`A\rightarrow B=\{(1,7),(2,6),(3,8),(4,5)\}`

(15)
`A\rightarrow B=\{(1,7),(2,5),(3,6),(4,8)\}`

(16)
`A\rightarrow B=\{(1,7),(2,5),(3,8),(4,6)\}`

(17)
`A\rightarrow B=\{(1,7),(2,8),(3,6),(4,5)\}`

(18)
`A\rightarrow B=\{(1,7),(2,8),(3,5),(4,6)\}`

(19)
`A\rightarrow B=\{(1,8),(2,6),(3,7),(4,5)\}`

(20)
`A\rightarrow B=\{(1,8),(2,6),(3,5),(4,7)\}`

(21)
`A\rightarrow B=\{(1,8),(2,7),(3,6),(4,5)\}`

(22)
`A\rightarrow B=\{(1,8),(2,7),(3,5),(4,6)\}`

(23)
`A\rightarrow B=\{(1,8),(2,5),(3,6),(4,7)\}`

(24)
`A\rightarrow B=\{(1,8),(2,5),(3,7),(4,6)\}`