For the alphabet $\sum = \{0,1\}, let A,B,C \subseteq \sum^*$ be the languages below.
$i. A = \{1, 0, 00, 11, 000, 111, 0000, 1111\}$
$ii. B = \{w \in \sum^*|||w|| \ge 2 \}$
$ii. C = \{w \in \sum^*|||w|| \le 2 \}$
Note: ||w|| is the length of the word w.
Determine the following languages of $\sum^*$:
a. $A \cap B$
b. $A \cap C$
c. $B \cup C$
d. $A \Delta B$
For $A \cap B$ I have: $w \in \sum^*$ such that $w \ge 2$, thus $A \cap B = 00, 11, 000, 111, 0000, 1111$. I am not sure if I am doing this right so any help is appreciated.
I completed the answers for $A \cap C$ and $B \cup C$ in a similar manner so I won't put the workings here.
I am completely stuck on $A \Delta B$. I am not sure how to use this character $\Delta$ in this exercise so any help is appreciated.
Thanks,
Tony
You answered $A \cap B$ correctly. The $\triangle$ operator on sets is called the symmetric difference operator. You only keep elements in exactly one of the sets. When dealing with binary values, the symmetric difference is analogous to the exclusive-or operator (or addition modulo 2).
Given disjoint sets, symmetric difference is equivalent to union, as neither sets share common elements, so nothing is removed.