Prove Identities of Sets Hello! (Wave)

In summary, you have successfully proven the following identities regarding set operations: $A \cap A^c= \varnothing, A \cup A^c=U, (A^c)^c=A, (A \cap B)^c=A^c \cup B^c, (A \cup B)^c=A^c \cap B^c, A \setminus B=A \cap B^c$.
  • #1
evinda
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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:



    • 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)
 
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  • #2


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!
 

Related to Prove Identities of Sets Hello! (Wave)

1. What is the purpose of proving identities of sets?

The purpose of proving identities of sets is to show that two sets are equal by demonstrating that every element in one set is also in the other set, and vice versa. This helps to establish a logical relationship between sets and can be useful in various mathematical and scientific applications.

2. How do you prove identities of sets?

To prove identities of sets, you must show that each element in one set is also in the other set, and vice versa. This can be done through various methods such as using Venn diagrams, set builder notation, or algebraic manipulation. It is important to follow proper mathematical logic and notation when proving identities of sets.

3. What is the difference between proving identities of sets and proving equalities of sets?

The main difference between proving identities of sets and proving equalities of sets is that proving identities involves showing that two sets are equal, while proving equalities involves showing that two sets have the same elements but may not necessarily be equal. Proving identities requires stricter criteria and must be true for all elements in the sets, whereas proving equalities may only need to be true for a subset of the elements.

4. Can you give an example of proving identities of sets?

Sure, an example of proving identities of sets would be to show that the set of even numbers (2, 4, 6, 8, etc.) is equal to the set of numbers divisible by 2 (2, 4, 6, 8, etc.). This can be proven by showing that every element in the set of even numbers is also in the set of numbers divisible by 2, and vice versa.

5. Why is proving identities of sets important in science?

Proving identities of sets is important in science because it allows us to establish a logical relationship between sets and make accurate conclusions based on the equality of sets. This can be useful in various fields of science, such as statistics, genetics, and computer science, where sets are commonly used to represent and analyze data.

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