# Inequality relations used in Multiple Sums

I am reading Chapter 2 of Concrete Mathematics, and have some trouble understanding the rationale behind two identities used for simplification.

1) [1<=j<=k<=n]+[1<=k<=j<=n] = [1<=j,k<=n]+[1<=j=k<=n]
2) [1<=j<k<=n]+[1<=k<j<=n] = [1<=j,k<=n]-[1<=j=k<=n]


Does anyone have any insight on how to prove this, intuition for this or way to think about it. I took some examples of 3X3 matrix and laid out the terms, the relation holds but I don't have an intuition of why it does. Any help is greatly appreciated. Thanks!

Here's how to think about it: it all starts with $$[1\le j,k\le n]$$. Subtract the other term from both sides to get $$[1\le j,k\le n] = [1\le j\le k\le n]+[1\le k\le j\le n] - [1\le j=k\le n]$$
If $$j,k$$ are not both int the interval $$[1,n]$$, then both sides are just zero. If instead they are, then the LHS is $$1$$. Now, there are three possibilities: $$j, $$j>k$$, $$j=k$$. If $$j, then the first term on the right is $$1$$, while the others are zero. If $$j>k$$, then the second term on the right is $$1$$, while the others are $$0$$. Lastly, if $$j=k$$, then the first two terms on the right are both $$1$$. However, we then add $$-1$$ from the last term, to get $$1$$ again, via $$1+1-1$$.