Explain how there exists one pair of infinite sets that are not equivalent?

Question

Basically, demonstrate the existence of two infinite sets that cannot be placed into a one-to-one corrospondence with each other....

Answer 1

The set of all natural numbers, and the set of all fractions are equivalent. However, the set of all decimals is not. For a good proof of this, look up George Cantor in wikipedia. The set of all irrational numbers is another example.

Answer 2

Let X be any infinite set (the set of natural numbers will do), and P(X) be the powerset of X -- i.e. the set of all subsets of X. Then there is no bijection between X and P(X). For suppose the contrary -- let f:X → P(X) be such a bijection. Then define S={x∈X: x∉f(x)}. Now, either f⁻¹(S)∈S or f⁻¹(S)∉S. If f⁻¹(S)∈S, then by the definition of S, we have f⁻¹(S) ∉ f(f⁻¹(S)) = S, a contradiction. However, if f⁻¹(S)∉S = f(f⁻¹(S)), then by the definition of S, f⁻¹(S)∈S, which is also a contradiction. Thus in either case we get a contradiction, so the proposed bijection cannot exist. Q.E.D.

Answer 3

Here's the simplest proof I know.

For one set take N, the natural numbers. For the other, take S, the set of all infinite sequences of natural numbers.

Suppose there is a map M:N-->S that is onto. Then construct an element D of S whose first element is greater by 1 than the first element of M(1), whose second element is greater by 1 than the second element of M(2), and so on.

Then there is no n in N such that D = M(n), because in fact D differs from M(n) in the nth position.

Contradiction. Q.E.D.

Answer 4

The integers and rationals are countable sets
but the set of real numbers is uncountable.
This was proved by G. Cantor in the late 19th
century.

Answer 5

by ability of "infiinet regress", i wager you propose to assert Who change into before God or Who created God. the answer is that no one created God. he's the Alpha and the Omega, the start and the end. It is smart to me, because if someone created God, then that someone must be better effective than God. if that is so, then God couldn't be God yet someone a lot less. and that is contradictory to our idea of God, this is that he's all-effective. So, that change into the lengthy thanks to assert that you're properly that countless regress (or organic regulation) would not stick with to the supernatural. The universe is "in basic terms" some billion years previous. It not in any respect existed perpetually. a minimum of, from a human perspective. To God, although, all available universes have continuously existed with him; this is, perpetually. countless regress would not clarify the age of the universe. this is a good judgment that feeds lower back on itself (this is yet all over again period for recursive), to at least some thing this is illogical or absurd.

Answer 6

1) an infinite set of whole numbers

2) an infinite set of fractions that are not equal to whole numbers

Answer 7

in mr dorffs class, eh??