In one of my previous question , I came across the idea $x \leq y$ is a disjunction of two mathematical statements - $x < y$ and $x = y$.
But according to the truth table for disjunction , It's also true in the condition when both the statements are true , but in case $x \leq y$ , both $x \leq y$ and $x = y$ cannot be simultaneously be true because of law of trichotomy .
So will it still be disjunction if an entire condition is invalidated , because then logically the statement will be true only in 2 conditions and false is 1 condition - consideration for the 4th condition (where both the statements are true) is invalid and so it's not even in the truth table ? Is it still Disjunction ?
Yes. We say that such statements as "if $ x \lt y \land x= y$ then $x \le y$" are "vacuously true": as the "if" clause can never be satisfied, the statement overall is always true. More true statements are "if $ x \lt y \land x= y$ then $1=0 $" or "if $2=1$ then $1=0$".
We could instead use an "xor" gate, which returns true iff exactly one of the two inputs is true, but we may as well use disjunction; the conditions are equivalent.