I have two operators! They are very close, brothers, actually! Though they are like each other's negative.They are named $\mathfrak{C}^+$ and $\mathfrak{C}^-$.
No one knows what's going on in their heads. It is now crucial! These two operators have concocted a plan to overthrow the common core! The common core government, however, have kidnapped $\mathfrak{C}^+$ and $\mathfrak{C}^-$!
This is a problem. Their mother, $\mathfrak{C}^\pm$, is worried to death. Their father, $\Gamma$, first needs to overthrow the tyrannous common core, but, to do so, he needs his children's plans! If only... if only we could find out what is going on in their heads!!
But news! Their father received a ransom from the common core; in exchange for their children, $\mathfrak{C}^\pm$ and $\Gamma$ must pay \$$g_{64}$! The ransom was penned in the two children's writing, and hurriedly scribbled in the margins is some I/Os. This is the key! Since $\mathfrak{C}^\pm$ can tell what her children are talking about if they tell her, she can figure out what their plan is!
So everything seems all figured out. Except for one thing. No one can find what functions $\mathfrak{C}^+$ and $\mathfrak{C}^-$ are. This is where you come in.
Given the I/O sheet, find out $\mathfrak{C}^+$'s function rule is! From there, the mother will decipher the hidden clues in the message. (The other child did not pen the letter)
Objective: Find the function rule for $\mathfrak{C}^+$. That is all.
Here is the sheet: $$X\subseteq Y \implies \mathfrak{C}^+(X,Y)=Y$$
$$\mathfrak{C}^+(1,2) = \sim6$$ $$\mathfrak{C}^+(1,5) = 9$$ $$\mathfrak{C}^+(1,8) = 8$$ $$\mathfrak{C}^+(2,3) = \sim6$$ $$\mathfrak{C}^+(2,7) = \sim6$$ $$\forall x\in\{0,4,5,6,8,9\}[\mathfrak{C}^+(2,x)=8]$$ $$\mathfrak{C}^+(4,7) = 9 \setminus \{\bot,\bot,\bot,\bot,\top,\bot,\top\}$$
Hints
Ignore the story. It's just for colour.
z1110111
o0010010
t1011101
t1011011
f0111010
f1101011
s1101111
s1010010
e1111111
n1111011
The numbers and letters are sets, which contain either $\top$s or $\bot$s; $\{\bot\}\subseteq\{\top\}$; think about operations of these symbols corresponding to numbers.
time is like a drug (LSD), except without the ex's and PT's (switch 'em)
I believe I have put up sufficient information. Please inform me if this belief is incorrect.
Good luck!!!
Edit: An apology
A key part of the riddle failed to display in mathjax; I have denoted some numbers with a tilde, but did not display. Sorry!
;)
. It is related to $\subseteq$ in the context of sets, but its not the same. $\endgroup$