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.

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 ].

"True and False?If the product A × B..." solved and explained

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Statistics and Probability
College Statistics
False Positive

posader86

Answered question

2022-07-20

True and False?
If the product

A \times B

of two sets

A

and

B

is the empty set , then both

A

and

B

have to be empty set.

Answer & Explanation

Octavio Barr

Beginner2022-07-21Added 11 answers

False, one of them being empty suffices. This one is at the heart of possible confusion about

k \times l

matrices with

k = 0

l = 0

(possibly both). You need to distinguish various types of empty matrices if you want to define multiplication correctly (a

n \times 0

matrix multiplied by a

0 \times m

matrix gives a non-empty, though zero,

n \times m

matrix), but you cannot make this distinction if you define matrices as a map on

[k] \times [l]

assigning entries to positions, because

[k] \times [0] = \emptyset = [0] \times [l]

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