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Question: If \(A = \left\\{ {1,2,3,4} \right\\}\) and \(B = \left\\{ {a,b,c,d} \right\\}\). Define any four bi...

If A = \left\\{ {1,2,3,4} \right\\} and B = \left\\{ {a,b,c,d} \right\\}. Define any four bijections from AA to BB. Also give their inverse functions.

Explanation

Solution

Hint: Try to make functions from given sets.

We know, A = \left\\{ {1,2,3,4} \right\\} and B = \left\\{ {a,b,c,d} \right\\}
A function from AA to BB is said to be bijection if it is one-one and onto. This means different elements of AA has different images in BB.
Also, each element of BB has preimage in AA.
Let f1,f2,f3{f_1},{f_2},{f_3}andf4{f_4}are the functions from AA to BB.

{f_1} = \left\\{ {\left( {1,a} \right),\left( {2,b} \right),\left( {3,c} \right),\left( {4,d} \right)} \right\\} \\\ {f_2} = \left\\{ {\left( {1,b} \right),\left( {2,c} \right),\left( {3,d} \right),\left( {4,a} \right)} \right\\} \\\ {f_3} = \left\\{ {\left( {1,c} \right),\left( {2,d} \right),\left( {3,a} \right),\left( {4,b} \right)} \right\\} \\\ {f_4} = \left\\{ {\left( {1,d} \right),\left( {2,a} \right),\left( {3,b} \right),\left( {4,c} \right)} \right\\} \\\

We can verify thatf1,f2,f3{f_1},{f_2},{f_3}andf4{f_4} are bijective from AA to BB.
Now,

{f_1}^{ - 1} = \left\\{ {\left( {a,1} \right),\left( {b,2} \right),\left( {c,3} \right),\left( {d,4} \right)} \right\\} \\\ {f_2}^{ - 1} = \left\\{ {\left( {b,1} \right),\left( {c,2} \right),\left( {d,3} \right),\left( {a,4} \right)} \right\\} \\\ {f_3}^{ - 1} = \left\\{ {\left( {c,1} \right),\left( {d,2} \right),\left( {a,3} \right),\left( {b,4} \right)} \right\\} \\\ {f_4}^{ - 1} = \left\\{ {\left( {d,1} \right),\left( {a,2} \right),\left( {b,3} \right),\left( {c,4} \right)} \right\\} \\\

Note: A bijection, bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.