threeder said:Homework Statement
Let [itex]X= \{a,b,c\} [/itex] and [itex]Y= \{d,e\}[/itex]. Write down and explicit bijection [tex]N_{|X×Y|} → X×Y[/tex]
The Attempt at a Solution
Well I came up with the easiest method, just giving one value to each member of [itex]N_{|X×Y|}[/itex] so I was just wondering whether there is another way of doing it not by brute force? :)
A bijection is a function between two sets where each element in one set is paired with exactly one element in the other set, and every element in the other set is also paired with exactly one element in the first set. In other words, a bijection is a one-to-one correspondence between two sets.
Writing down an explicit bijection is important because it allows us to clearly define the relationship between two sets and understand how each element in one set is mapped to an element in the other set. It also provides a concrete and verifiable proof of the existence of a bijection between two sets.
To write down an explicit bijection, you need to define a rule or function that maps each element in one set to a unique element in the other set. This can be done by using mathematical expressions, tables, or diagrams to illustrate the mapping between the two sets.
One example of an explicit bijection is the function f(x) = x, where x is a real number. This function maps each real number to itself, creating a one-to-one correspondence. Another example is the bijection between the set of natural numbers and the set of even numbers, where each natural number is paired with its corresponding even number.
To prove that a bijection is explicit, you need to show that each element in one set is mapped to a unique element in the other set, and every element in the other set is also mapped to a unique element in the first set. This can be done by using mathematical proofs or by showing examples that demonstrate the one-to-one correspondence between the two sets.