Confusion about logical constraints

Question

In my linear programming course, when discussing logical constraints, my notes read:

If item $i$ is selected, then item $j$ is not selected, and the constraint reads: $x_i + x_j \leq 1$.

I am confused about the $\leq$ sign, because say item $i$ is not selected, does that also mean that item $j$ is not selected?

Thanks.

Answer 1

This “conflict” constraint prevents selecting both items but allows either item or neither item to be selected. Of the four possibilities for $(x_i,x_j)$, only $(1,1)$ is excluded.