- #1

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The problem states:

“Determine whether the set of all functions from {0,1} to

**Z**is countable or uncountable.”

_{+}First, I took a look at what a few functions looked like. For instance, {(0,5),(1,7)} would constitute a function. In general, the set of all functions in this case would be the collection of all {(0,n),(1,m)}, where n and m are positive integers. It then hit me that I can construct a bijective correspondence between the set of functions from {0,1} to

**Z**and {0} X

_{+}**Z**X {1} X

_{+}**Z**by the mapping

_{+}((0,n),(1,m)) <-> (0,n,1,m).

The latter set is a finite product of countable sets, and therefore countable. The desired result will follow.Any input would be great! Thanks so much! =)