Hi do the following sets have the same cardinality, why or why not?
N= 1,2,3,4,5,6,7...
N-2= blank, blank, 3,4,5,6,7...
So basically the numbers from 3 to 7 match up in both sets, but in N-2 the first two are empty. I think they have the same cardinality but I don't know how to prove it. Please help this is due tommorow and it's my last problem.
To determine whether two sets have the same cardinality, we need to show that there exists a one-to-one correspondence, or a bijection, between the elements of the two sets. In other words, we need to demonstrate a way to pair up each element in one set with a unique element in the other set.
In this case, the set N contains all natural numbers starting from 1, while the set N-2 is missing the first two elements. The first step is to identify a function that matches the elements of N-2 with the elements of N.
To do this, we can define a function f, where f(x) = x + 2. This function takes an element x from N-2, adds 2 to it, and associates it with the element in N. For example, f(3) = 3 + 2 = 5, f(4) = 4 + 2 = 6, and so on.
Now, let's observe the pairing between the two sets:
N-2: blank blank 3 4 5 6 7 ...
N: 1 2 3 4 5 6 7 ...
As we can see, the element 3 in N-2 is paired with the element 3 in N, and so on. The blanks in N-2 do not pose a problem since we are only concerned with the elements that are present.
Since we can establish a clear one-to-one correspondence between the elements of N-2 and N using the function f, we can conclude that the sets N and N-2 have the same cardinality. Therefore, they have an equal number of elements.
In summary, the sets N and N-2 have the same cardinality because there exists a function f(x) = x + 2 that pairs each element in N-2 with a unique element in N.