Although the set of real numbers and the set of natural numbers each contain an infinite numbers of elements, the set of reals has a higher cardinality than the set naturals. Why is that?

I tell non-math I'm studying linear algebra and they say: "that sounds like something I took in sixth grade."
So why call it linear algebra? It is supposed to be in contrast to "abstract algebra" ?
Thanks

I need to show that [latex] f: (0,1) \rightarrow R[/latex] is 1-1, where [latex] f(x) [/latex] is given by [latex] (2x-1)/(x^2-x) [/latex].
My attempt: suppose:[latex] f(x) = f(y)[/latex]
Then: [latex] (2x-1)/(x^2-x) [/latex] = [latex] (2y-1)/(y^2-y) [/latex].
[latex]y(y-1)(2x-1) = (x(x-1))(2y-1)[/latex]
Reducing this leads to:
[latex]2xy^2 - y^2 + y = 2x^2 - x^2 + x[/latex]
Of course, I need to show that [latex]x = y[/latex] but I'm not sure how to reduce this equality any further. Any ideas?
Thanks

Hello everyone,
I want to make sure I understand the Cartesian product of two sets.
Let A = {1, 2}
B= {3, 4}
Then A X B = {(1,3),(1,4),(2,3),(2,4)}
Is that correct?
Thanks

Hello everyone,
For as long as I have known about the constant [latex]e[/latex], I have been in awe of its many uses and at times strange properties. Now I know its definition as a limit and as a series, but I have never quite understood just what makes [latex]e[/latex] so special, beyond the fact that it helps us solve problems. So my question is: what makes [latex]e[/latex] so significant to the overall study of mathematics?
I hope that makes sense. I look forward to others perspective on this.
Thanks

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