What does \sum\limits_{i\neq j}^N a_i a_j mean in summation notation?

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The notation \sum\limits_{i\neq j}^N a_i a_j represents the sum of products of elements a_i and a_j for all pairs where i is not equal to j. It is equivalent to the double summation \sum\limits_{i}^N \sum\limits_{j}^N a_i a_j, excluding cases where j equals i. For example, with N = 3, the summation includes terms like a1a2, a1a3, and a2a3, while omitting a1a1, a2a2, and a3a3. This notation is commonly used in mathematical contexts, though it can sometimes be misapplied in set theory discussions. Understanding this notation is essential for correctly interpreting summations in mathematical texts.
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Hi

I have a textbook which uses the notation:

\sum\limits_{i\neq j}^N a_i a_j

I can't find anywhere what this actually means. Is it equivalent to:

\sum\limits_{i}^N \sum\limits_{j}^N a_i a_j

where j can't equal i?

Thanks.
 
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It's just the sum of all ##a_{i}a_{j}## when i does not equal j.

You can put pretty much any condition you want in that space, though I see it abused more often for unions and intersections of sets.
 
SergeantAngle said:
Hi

I have a textbook which uses the notation:

\sum\limits_{i\neq j}^N a_i a_j

I can't find anywhere what this actually means. Is it equivalent to:

\sum\limits_{i}^N \sum\limits_{j}^N a_i a_j

where j can't equal i?

Thanks.

Yes, pretty much. For example, if N = 3, and both indexes start at 1, then the summation expands to a1a2 + a1a3 + a2a1 + a1a3 + a2a3 + a3a2. The terms that are omitted are a12, a22, and a32.
 
Okay, thanks.
 
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