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# Common Misconceptions (Algebra)

Arm yourself with the tools to be the king or queen of heady mathematical debates, like the age old question of whether 0.999.... = 1.

*cardinality* if there is a one-to-one correspondence between all their elements. In the picture above, there is a one-to-one correspondence between the blue dots and the red dots in Figure 1, but there is not a one-to-one correspondence in Figures 2 and 3.
Which of the sets below have the same cardinality?

True or False?

There is a one-to-one correspondence between the set \(\mathbb{N} = \{0, 1, 2, 3, \ldots \}\) of non-negative and the set \(\mathbb{Z} = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots \}\) of integers.

*infinitely many* travelers (Traveler 1, Traveler 2, Traveler 3, ...) wander in, each looking for a room. The manager wants to figure out a way to accomodate all the travelers without moving anyone else out. Is this possible?

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