Proving the Equality of Finite Cartesian/Cross Products for Sets A and B

In summary, the problem is to prove that for finite sets A and B, the cardinality of their cartesian product is equal to the product of their individual cardinalities. The attempt at a solution involved using an example with A = {1,2} and B = {3,4}, but the resulting cross product was incorrect. The correct cross product is AxB = {{1,3}, {1,4}, {2,3}, {2,4}}, which has a cardinality of 4. This is equal to the product of the cardinalities of A and B, which is also 4. Therefore, m(AxB) = m(A)m(B).
  • #1
kmikias
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Homework Statement



show for finite sets A,B,that m(A x B ) = m(A)m(B).

Homework Equations



I don't see relevant equation but we can treat to like cartesian/cross product.


The Attempt at a Solution



I tried to think it as (x,y) for m(A x B ) . if we let A = { 1,2} and B= {3,4} then those product will be {1,3},{2,4},{1,4},{1,4} isn't it? but then i am confused with m(A)m(B) because it seems like multiplication .

I need some hint to prove this statement/question.
thank you.
 
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  • #2
m(A)m(B) is multiplication. In your example you listed the elements of the cross product. But you got it wrong. AxB={{1,3},{1,4},{2,3},{2,4}}. There are 4 elements in there. So m(AxB)=4. m(A)=2 and m(B)=2. So m(A)m(B)=2*2=4. So m(AxB)=m(A)m(B).
 

1. What is a Cartesian product?

A Cartesian product is a mathematical operation that combines two sets to form a new set, where each element of the new set is a pair consisting of an element from each of the original sets. This operation is commonly used in set theory and in vector algebra.

2. What is a cross product?

A cross product is a mathematical operation that results in a vector that is perpendicular to both of the input vectors. The magnitude of the resulting vector is equal to the product of the magnitudes of the two input vectors multiplied by the sine of the angle between them. This operation is commonly used in vector algebra, particularly in three-dimensional space.

3. How do you prove the properties of the Cartesian product?

The properties of the Cartesian product can be proven using mathematical proofs, which involve logical reasoning and rigorous mathematical steps. These proofs typically start with the definition of the Cartesian product and use mathematical properties and theorems to show that the properties hold true. These proofs require a strong understanding of mathematical concepts and logical thinking.

4. What is the difference between the Cartesian product and the cross product?

The main difference between the Cartesian product and the cross product is that the Cartesian product combines two sets to form a new set, while the cross product combines two vectors to form a new vector. Additionally, the resulting elements of the Cartesian product are pairs, while the resulting vector of the cross product is perpendicular to the input vectors.

5. How are the properties of the Cartesian product and the cross product related?

The properties of the Cartesian product and the cross product are related in that they both involve combinations of elements from two sets or vectors. However, the properties themselves are different as they pertain to different mathematical operations. Additionally, some properties may hold true for one operation but not the other, so it is important to understand the specific properties of each operation when proving them.

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