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Given a 2d plane. With a coordinate system, you can identify an infinite amount of points. That is, the set S of points that lie on the plane has an infinite number of elements.

Now say you only consider the first quadrant of the plane (taking the usual x, y axes as the coordinate system). The set of points Q that lie on the first quadrant still has an infinite number of elements.

However, intuitively speaking, |S| > |Q|.

The same scenario can be done with a number line and the positive number line.

I don't know much about infinities mathematically. Is this mathematical, or just misleading intuition?

Saksham
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    It is misleading intuition. If there is a bijection between $A$ and $B$ then $|A|=|B|.$ A simpler example is $A=\Bbb N$ and $B=2\Bbb N.$ Related: A plane has the same cardinality as a half-plane. – Anne Bauval Jul 11 '23 at 10:32
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    There are indeed different types of infinity, some infinities are bigger than others. For example, there are more real numbers than natural numbers. However, the two sets you asked about have the same cardinality. Look up for "cardinality of a set". – Mark Jul 11 '23 at 10:33
  • This is far too broad of a question. TLDR is that yes, some infinite sets are bigger than others, but just because one set is a subset of another does not mean it is smaller. For example, the set of all integers and the set of all rational numbers are equipotent (equal in "size"). See https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel here for an introduction. – 5xum Jul 11 '23 at 10:34
  • If you have an intuition try to formalize it. First it is bigger in what sense? Like you said $|S|=|Q|$. What does cardinality mean? Try to understand the definition cardinality. (2) Then ask yourself how do you normally count? And why should we accept that definition of cardinality?
    • – PNDas Jul 11 '23 at 10:38
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    It is actually very easy to make a bijection of $(0, +\infty)$ to the entire $\mathbb R$: the logarithm function ($x\mapsto\log x$) does it outright for you. Its inverse map is the exponential function $x\mapsto e^x$, which maps $\mathbb R$ to $(0,+\infty)$. You may want to use these functions (twice) to make a bijection between $S$ and $Q$. –  Jul 11 '23 at 10:40
  • Thank you for your responses. I was assuming that if A is a subset of B, they cannot be of the same 'size'. After some research, with the case of infinite sets, it doesn't seem to apply. – Saksham Jul 11 '23 at 11:31
  • @Saksham In fact, that is one of the characteristics of infinite sets! In other words, a set $A$ is infinite if and only if there exists some proper subset $B\subset A$ (proper meaning $B\neq A$) such that $|A|=|B|$ (i.e., that the two are of the same size). – 5xum Jul 11 '23 at 11:37