# How to interpret this sample space set?

by Ben Watson   Last Updated October 18, 2019 09:20 AM

As part of one of my Random Signal Theory assignments I came across the sample space of random variables X and Y represented by the set:

$$S_{X,Y} = \{(x_i,y_j):(i,j),i,j=1,2\}$$

How would I interpret this? What I think it means is: "Sample space $$S_{X,Y}$$ contains $$x_i$$ and $$y_j$$ such that $$i$$ and $$j$$ can take the values $$1$$ or $$2$$"

Which would result in the sample space containing 4 variables: $$x_1,x_2,y_1,y_2$$

However I am mostly going on intuition and lack a true understanding of how exactly to construct and interpret this sort of notation.

Another option in my mind is that it is somehow specifies that: $$x_1=1, x_2=2, y_1=1, y_2=2$$ Which would fit the context of the problem, but again I have no idea how to derive this meaning from the above statement.

Can someone please explain exactly what this does mean and/or point me in the direction of learning resources that would teach me the basics of understanding this set notation?

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I would interpret this as the set of pairs $$(x_i, y_j)$$ with $$i,j = 1, 2$$. Explicitly, the set $$S_{X,Y} = \{(x_1, y_1), (x_2, y_1), (x_1, y_2), (x_2, y_2)\}.$$

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