Respuesta :
Answer:
The list of bijections from A into B are shown below.
Step-by-step explanation:
A function f is called one-to-one or injective, if and only if
[tex]f(x)=f(y)\Rightarrow x = y[/tex]
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) [tex]A\rightarrow B=\{(q,2),(r,3),(s,4)\}[/tex]
(2) [tex]A\rightarrow B=\{(q,2),(r,4),(s,3)\}[/tex]
(3) [tex]A\rightarrow B=\{(q,3),(r,2),(s,4)\}[/tex]
(4) [tex]A\rightarrow B=\{(q,3),(r,4),(s,2)\}[/tex]
(5) [tex]A\rightarrow B=\{(q,4),(r,2),(s,3)\}[/tex]
(6) [tex]A\rightarrow B=\{(q,4),(r,3),(s,2)\}[/tex]
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) [tex]A\rightarrow B=\{(1,5),(2,6),(3,7),(4,8)\}[/tex]
(2) [tex]A\rightarrow B=\{(1,5),(2,6),(3,8),(4,7)\}[/tex]
(3) [tex]A\rightarrow B=\{(1,5),(2,7),(3,6),(4,8)\}[/tex]
(4) [tex]A\rightarrow B=\{(1,5),(2,7),(3,8),(4,6)\}[/tex]
(5) [tex]A\rightarrow B=\{(1,5),(2,8),(3,6),(4,7)\}[/tex]
(6) [tex]A\rightarrow B=\{(1,5),(2,8),(3,7),(4,6)\}[/tex]
(7) [tex]A\rightarrow B=\{(1,6),(2,5),(3,7),(4,8)\}[/tex]
(8) [tex]A\rightarrow B=\{(1,6),(2,5),(3,8),(4,7)\}[/tex]
(9) [tex]A\rightarrow B=\{(1,6),(2,7),(3,5),(4,8)\}[/tex]
(10) [tex]A\rightarrow B=\{(1,6),(2,7),(3,8),(4,5)\}[/tex]
(11) [tex]A\rightarrow B=\{(1,6),(2,8),(3,5),(4,7)\}[/tex]
(12) [tex]A\rightarrow B=\{(1,6),(2,8),(3,7),(4,5)\}[/tex]
(13) [tex]A\rightarrow B=\{(1,7),(2,6),(3,5),(4,8)\}[/tex]
(14) [tex]A\rightarrow B=\{(1,7),(2,6),(3,8),(4,5)\}[/tex]
(15) [tex]A\rightarrow B=\{(1,7),(2,5),(3,6),(4,8)\}[/tex]
(16) [tex]A\rightarrow B=\{(1,7),(2,5),(3,8),(4,6)\}[/tex]
(17) [tex]A\rightarrow B=\{(1,7),(2,8),(3,6),(4,5)\}[/tex]
(18) [tex]A\rightarrow B=\{(1,7),(2,8),(3,5),(4,6)\}[/tex]
(19) [tex]A\rightarrow B=\{(1,8),(2,6),(3,7),(4,5)\}[/tex]
(20) [tex]A\rightarrow B=\{(1,8),(2,6),(3,5),(4,7)\}[/tex]
(21) [tex]A\rightarrow B=\{(1,8),(2,7),(3,6),(4,5)\}[/tex]
(22) [tex]A\rightarrow B=\{(1,8),(2,7),(3,5),(4,6)\}[/tex]
(23) [tex]A\rightarrow B=\{(1,8),(2,5),(3,6),(4,7)\}[/tex]
(24) [tex]A\rightarrow B=\{(1,8),(2,5),(3,7),(4,6)\}[/tex]
