Let A and B be two subsets of some universal set. Prove that....

In summary, the conversation discusses proving the equality of two subsets, A and B, of a universal set. The proof involves showing that if the complement of their union is equal to the union of their complements, then A must be equal to B. This is done by assuming an element x is in A and showing that it must also be in B, and vice versa, demonstrating that the two subsets are equal. The conversation also suggests an alternative approach using symmetry and commutativity.
  • #1
KOO
19
0
**Let A and B be two subsets of some universal set.
Prove that if $(A\cup B)^c$ = $A^c$ U $B^c$, then A = B.**Attempt:

Let $x\in A$. Then $x\in A\cup B$, so $x\notin(A\cup B)^c$. By hypothesis $(A\cup B)^c=A^c\cup B^c$, so $x\notin A^c\cup B^c$. In particular, then, $x\notin B^c$, and therefore $x\in B$. Since $x$ was an arbitrary element of $A$, this shows that $A\subseteq B$.

How do we show $B\subseteq A$?
 
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  • #2
KOO said:
**Let A and B be two subsets of some universal set.
Prove that if $(A\cup B)^c$ = $A^c$ U $B^c$, then A = B.**Attempt:

Let $x\in A$. Then $x\in A\cup B$, so $x\notin(A\cup B)^c$. By hypothesis $(A\cup B)^c=A^c\cup B^c$, so $x\notin A^c\cup B^c$. In particular, then, $x\notin B^c$, and therefore $x\in B$. Since $x$ was an arbitrary element of $A$, this shows that $A\subseteq B$.

How do we show $B\subseteq A$?

Start with "Let $x\in B$."
Then continue with the same argument you have - just with $A$ and $B$ swapped around...
 
  • #3
Or...just note that both expressions are symmetric in A and B, and union is commutative...
 

Related to Let A and B be two subsets of some universal set. Prove that....

1. What is a universal set?

A universal set is a set that contains all the elements that are being discussed or considered in a particular context. It serves as a reference point for other sets in a given scenario.

2. What are subsets?

Subsets are sets that contain elements that are also found in another set. They are smaller than the original set and may or may not have the same number of elements.

3. How do you prove that A and B are subsets of a universal set?

To prove that A and B are subsets of a universal set, you need to show that all the elements in A and B are also found in the universal set. This can be done by listing out all the elements in A and B and comparing them to the elements in the universal set.

4. What is the importance of proving that A and B are subsets of a universal set?

Proving that A and B are subsets of a universal set is important because it helps in understanding the relationships between sets and their elements. It also allows for the use of mathematical operations such as union, intersection, and complement to be performed on the sets.

5. Are there any specific techniques or methods for proving that A and B are subsets of a universal set?

Yes, there are various techniques and methods that can be used to prove that A and B are subsets of a universal set. These include direct proof, contrapositive proof, proof by contradiction, and mathematical induction. The choice of method depends on the specific context and the properties of the sets being considered.

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