Conjunction of Sets: x,y in Rm+n vs Vector z in R(m+n)

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In summary, if x is an element of set A which is a subset of real numbers with n dimensions, and y is an element of set B which is a subset of real numbers with m dimensions, then the Cartesian product of A and B is a subset of real numbers with m+n dimensions. There is no difference in connotation between (x,y) being an element of R with m+n dimensions and a vector z being in R with (m+n) dimensions. This is because the notation is isomorphic, meaning there is a one-to-one correspondence between z and (x,y). The notation R^{m+n} and R^{(m+n)} are not distinguished in this case.
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If x [tex]\epsilon[/tex] A [tex]\subset[/tex] R[tex]\overline{}n[/tex] and y [tex]\epsilon[/tex] B [tex]\subset[/tex] R[tex]\overline{}m[/tex],

then A X B [tex]\subset[/tex] R[tex]\overline{}m+n[/tex].

Is there any difference between the connotation of (x,y) [tex]\epsilon[/tex] R[tex]\overline{}m+n[/tex] and a vector z in R[tex]\overline{}(m+n)[/tex]?

thanks!

(can't get the (m+n) above the line yet)
 
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the short answer is no. But this is because when the notation is like that, it is clearly isomorphic. That is to every z there is a unique (x,y) and the other way around.

Don't know why you distinguish between [itex]R^{m+n}[/itex] and [itex]R^{(m+n)}[/itex], have never seen that notation.
 

Related to Conjunction of Sets: x,y in Rm+n vs Vector z in R(m+n)

1. What is the difference between "x,y in Rm+n" and "Vector z in R(m+n)"?

The notation "x,y in Rm+n" refers to a conjunction of two sets, where both x and y are elements of the set Rm+n. On the other hand, "Vector z in R(m+n)" refers to a single vector, z, that is an element of the set R(m+n). In other words, "x,y in Rm+n" refers to two separate elements that both belong to the set Rm+n, while "Vector z in R(m+n)" refers to a single element that belongs to the set R(m+n).

2. Can you provide an example of "x,y in Rm+n" and "Vector z in R(m+n)"?

Sure, let's say we have the sets R2+3 = {(x,y) | x,y in R and x+y=2} and R(2+3) = {(z1,z2,z3) | z1,z2,z3 in R and z1+z2+z3=5}. An example of "x,y in R2+3" would be (1,1) since 1+1=2 and both 1 and 1 are elements of the set R. An example of "Vector z in R(2+3)" would be (1,2,2) since 1+2+2=5 and all three elements (1,2,2) are in the set R.

3. How do you determine if a given element is in the set Rm+n?

To determine if a given element is in the set Rm+n, you need to check if it satisfies the conditions set for the set. For example, if we have the set R2+3 = {(x,y) | x,y in R and x+y=2}, to see if the element (1,1) is in the set, we would plug in x=1 and y=1 into the condition x+y=2. Since 1+1=2, (1,1) satisfies the condition and is therefore in the set R2+3.

4. Can you explain the significance of the notation "Rm+n"?

The notation "Rm+n" represents a set of vectors in a Euclidean space, where the vectors have m+n dimensions. The letter "R" stands for the set of real numbers, and the subscript "m+n" indicates the number of dimensions of the vectors in the set. For example, R2+3 would represent a set of vectors with 2+3=5 dimensions, where each element in the vector is a real number.

5. What are some applications of "Conjunction of Sets: x,y in Rm+n vs Vector z in R(m+n)"?

The concept of "Conjunction of Sets: x,y in Rm+n vs Vector z in R(m+n)" is commonly used in linear algebra, specifically in the study of vector spaces and subspaces. It is also used in fields such as physics, engineering, and computer graphics, where vectors and vector spaces play a crucial role in representing physical quantities and systems. In addition, this concept has applications in data analysis and machine learning, where vectors are used to represent data points and vector spaces are used to model relationships between data points.

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