Question

Let S be the set of real numbers between 0 and 1, inclusive; i.e. S = [0, 1]. Let T be the set of real numbers between 1 and 3 inclusive (i.e. T = [1, 3]). Show that S and T have the same cardinality.

Answer #1

Two sets A and B have the same cardinality iff there exists a bijection from one onto the other.

The solution has been obtained by exhibiting an explicit bijection from S onto T. A pictorial representation showing such a bijection has been provided. Loosely speaking, every point on the interval [0,1] is related to a unique point on [1,3] via the bijection. The endpoints of S are mapped onto the corresponding endpoints of T.

15.)
a) Show that the real numbers between 0 and 1 have the same
cardinality as the real numbers between 0 and pi/2. (Hint: Find a
simple bijection from one set to the other.)
b) Show that the real numbers between 0 and pi/2 have the same
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function whose graph goes from 0 to positive infinity as x goes
from 0 to pi/2?)
c) Use parts a and b to...

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