Prove Identities of Sets Hello! (Wave)

[evinda]

Hello! (Wave)

Let $U$ be a set and $A,B$ subsets of $U$.

I want to prove the following identities:

1. $A \cap A^c= \varnothing, A \cup A^c=U$
2. $(A^c)^c=A$
3. $(A \cap B)^c=A^c \cup B^c$
4. $(A \cup B)^c=A^c \cap B^c$
5. $A \setminus B=A \cap B^c$

That's what I have tried:

1. 
   
   • 
     Let $x \in A \cap A^c$.
     Then $x \in A \wedge x \in A^c \leftrightarrow x \in A \wedge x \in U \setminus A \leftrightarrow x \in A \wedge (x \in U \wedge x \notin A)$, that is a contradiction.
     
     So, there is no element $x$, such that $x \in A \cap A^c$, therefore $A \cap A^c=\varnothing$.
     $$$$
   • Let $x \in A \cup A^c$.
     Then $x \in A \lor x \in A^c \leftrightarrow x \in A \lor x \in U \setminus A \leftrightarrow x \in A \lor (x \in U \wedge x \notin A)$
     
     How can I continue?
   
   $$$$
2. Let $x \in (A^c)^c \leftrightarrow x \in U \setminus A^c \leftrightarrow x \in U \wedge x \notin A^c \leftrightarrow x \in A \cup A^c \wedge x \notin A^c \leftrightarrow (x \in A \lor x \in A^c) \wedge x \notin A^c \leftrightarrow x \in A$
   $$$$
3. Let $x \in (A \cap B)^c \leftrightarrow x \in U \setminus A \cap B \leftrightarrow x \in U \wedge x \notin A \cap B \leftrightarrow x \in U \wedge ((x \in A \wedge x \notin B) \lor (x \notin A \wedge x \in B)) \leftrightarrow (x \in U \wedge (x \in A \wedge x \notin B)) \lor (x \in U \wedge (x \notin A \wedge x \in B))$
   
   How can I continue?
   $$$$
4. Let $x \in (A \cup B)^c \leftrightarrow x \in U \setminus (A \cup B) \leftrightarrow x \in U \wedge x \notin A \cup B \leftrightarrow x \in U \wedge (x \notin A \wedge x \notin B) \leftrightarrow (x \in U \wedge x \notin A) \wedge (x \in U \wedge x \notin B) \leftrightarrow x \in A^c \wedge x \in B^c \leftrightarrow x \in A^c \cap B^c$
   $$$$
5. Let $x \in A \setminus B \leftrightarrow x \in A \wedge x \notin B \leftrightarrow x \in A \wedge x \in U \setminus B \leftrightarrow x \in A \wedge x \in B^c \leftrightarrow x \in A \cap B^c$

Could you tell me if that what I have tried is right? (Thinking)

 

[mmwave]

Yes, your reasoning and steps are correct. You have used the definition of set operations and the properties of logical connectives to prove each identity. Good job!