True and False? If the product AxB of two sets A and B is the empty set, then both A and B have to be empty set.

posader86 2022-07-20 Answered
True and False?
If the product A × B of two sets A and B is the empty set , then both A and B have to be empty set.
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Answers (1)

Octavio Barr
Answered 2022-07-21 Author has 11 answers
False, one of them being empty suffices. This one is at the heart of possible confusion about k × l matrices with k = 0 or l = 0 (possibly both). You need to distinguish various types of empty matrices if you want to define multiplication correctly (a n × 0 matrix multiplied by a 0 × m matrix gives a non-empty, though zero, n × m matrix), but you cannot make this distinction if you define matrices as a map on [ k ] × [ l ] assigning entries to positions, because [ k ] × [ 0 ] = = [ 0 ] × [ l ].
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