![]() ![]() There are many ways to talk about infinite sets. Note that “as many” is in quotes since these sets are infinite sets. There are “as many” prime numbers as there are natural numbers? ![]() There are “as many” positive integers as there are integers? (How can a set have the same cardinality as a subset of itself? :-) There are “as many” even numbers as there are odd numbers? We note that is a one-to-one function and is onto.Ĭan we say that ? Yes, in a sense they are both infinite!! So we can say !! There is a one to one correspondence between the set of all natural numbers and the set of all odd numbers. One-to-One Correspondences of Infinite Set How does the manager accommodate these infinitely many guests? How does the manager accommodate the new guests even if all rooms are full?Įach one of the infinitely many guests invites his/her friend to come and stay, leading to infinitely many more guests. ![]() Let us take, the set of all natural numbers.Ĭonsider a hotel with infinitely many rooms and all rooms are full.Īn important guest arrives at the hotel and needs a place to stay. We now note that the claim above breaks down for infinite sets. The last statement directly contradicts our assumption that is one-to-one. Therefore by pigeon-hole principle cannot be one-to-one. Is now a one-to-one and onto function from to. Similarly, we repeat this process to remove all elements from the co-domain that are not mapped to by to obtain a new co-domain. Therefore, can be written as a one-to-one function from (since nothing maps on to ). (b) Find the rank and nullity of the matrix $A$ in part (a).Let be a one-to-one function as above but not onto. Find a Basis of the Range, Rank, and Nullity of a Matrixįind a basis for the range $\calR(A)$ of $A$ that consists of columns of $A$. The matrix representation of the linear transformation $T$ is given by
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