show the set has cardinal number N0 by establishing a one-to-one correspondence

Question

show the set has cardinal number N0 by establishing a one-to-one correspondence between the set of counting numbers and the given set. Be sure to show the pairing of the general tens in the set.
{6,9,12,15,18,...} show the set has cardinal number N0 by establishing a one-to-one correspondence between the set of counting numbers and the given set. Be sure to show the pairing of the general tens in the set.
{6,9,12,15,18,...} show the set has cardinal number N0 by establishing a one-to-one correspondence between the set of counting numbers and the given set. Be sure to show the pairing of the general tens in the set.
{6,9,12,15,18,...}

Explanation / Answer

Consider the function f(x) = 3x + 3 defined over the set N -> {6,9,12,15,18,.....}.

Note that for any element x {6,9,12,15,18.....}, [(x - 3) / 3] {1,2,3,4,5....} or N.

If f(a) = f(b) then 3a + 3 = 3b + 3 => 3a = 3b => a = b and therefore, f is one-one.

Also if b N, b = 3[(b - 3) / 3] + 3 = f([(b - 3) / 3]) and therefore, f is onto.

Thus f is a bijection from N to {6,9,12,15,18......}.

Therefore, the set {6,9,12,15,18......} has the same cardinality as N.

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