कवल तक अवयव एक अनुक्रमक रूप मालख जा सक। प्रमेय 2.21. प्रत्येक अपरिमित गणनीय समुच्चय अपने किसी एक उचित उपसमुच्चय के साथ एकैकी संगति रखता है। Any denumerable set can be put into a one-one correspondence
Answers
Answer:
Preliminaries
N={1,2,3,4,...} is the set of Natural Numbers, also known as the Counting Numbers.(1)
N is an infinite set and is the same as Z+.
In this section, we will see how the the Natural Numbers are used as a standard to test if an infinite set is "countably infinite".
{1,2,3,...,n} is a FINITE set of natural numbers from 1 to n.(2)
Recall: a one-to-one correspondence between two sets is a bijection from one of those sets to the other. A bijection is a function that is one-to-one and onto.
Finite Sets
Finite sets are either empty or have n elements. If a set has n elements, there exists a one-to-one correspondence with the set of natural numbers, {1,2,3,...,n} where n∈N.
For example, {p,q,r} can be put into a one-to-one correspondence with {1,2,3} . One such function is p→1q→2r→3.
If set S has n elements, then |S|=n . Also |∅|=0.
Infinite Sets
An infinite set is a non-empty set which cannot be put into a one-to-one correspondence with {1,2,3,...,n} for any n∈N .
Cardinality
Cardinality is transitive (even for infinite sets).
For all sets A,B,C, if |A|=|B| and |B|=|C| then |A|=|C|(3)
Same Cardinality
If set A and set B have the same cardinality, then there is a one-to-one correspondence from set A to set B .
For a finite set, the cardinality of the set is the number of elements in the set.
Example 1
Consider sets P and Q . P={olives, mushrooms, broccoli, tomatoes} and Q={Jack, Queen, King, Ace}.
Since |P|=4 and |Q|=4 , they have the same cardinality and we can set up a one-to-one correspondence such as:
olives → Jack(4)
mushrooms → Ace(5)
broccoli → Queen(6)
tomatoes → King(7)
Theorem 1
An infinite set and one of its proper subsets could have the same cardinality
Yes
An infinite set that can be put into a one-to-one
correspondence with N is countably infinite. Finite sets and countably infinite are called countable. An infinite set that cannot be put into a one-to-one correspondence with N is uncountably infinite.