Logic of Set Operations: Proof

In summary, we used the definitions of set operations and the laws of logic to prove that the difference of the union and intersection of two sets is equal to the union of the differences of the two sets.
  • #1
solakis1
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Given:
\(\displaystyle x\in A\cap B\leftrightarrow x\in A\wedge x\in B\)
\(\displaystyle x\in A\cup B\leftrightarrow x\in A\vee x\in B\)
\(\displaystyle x\in A-B\leftrightarrow x\in A\wedge x\notin B\)
\(\displaystyle A=B\leftrightarrow(\forall x(x\in A\leftrightarrow x\in B))\)

Then prove using only the above and the laws of logic that:

\(\displaystyle (A\cup B)-(A\cap B)=(A-B)\cup(B-A)\)
 
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  • #2

To prove this statement, we will use the laws of logic and the given definitions of set operations.

First, we will rewrite the left side of the equation using the given definitions:
(A\cup B)-(A\cap B)=(x\in A\vee x\in B)-(x\in A\wedge x\in B)

Next, we will use the distributive law of logic to rewrite the right side of the equation:
(x\in A\vee x\in B)-(x\in A\wedge x\in B)=((x\in A\vee x\in B)\wedge \neg(x\in A\wedge x\in B))

Now, we will use the definition of set difference to rewrite the right side of the equation:
((x\in A\vee x\in B)\wedge \neg(x\in A\wedge x\in B))=((x\in A\wedge \neg(x\in A\wedge x\in B))\vee (x\in B\wedge \neg(x\in A\wedge x\in B)))

Using De Morgan's laws, we can simplify the right side of the equation:
((x\in A\wedge \neg(x\in A\wedge x\in B))\vee (x\in B\wedge \neg(x\in A\wedge x\in B)))=((x\in A\wedge \neg x\in B)\vee (x\in B\wedge \neg x\in A))

Finally, we can use the definition of set union to rewrite the right side of the equation:
((x\in A\wedge \neg x\in B)\vee (x\in B\wedge \neg x\in A))=((x\in A-B)\vee (x\in B-A))

Therefore, we have shown that (A\cup B)-(A\cap B)=(A-B)\cup(B-A) using only the given definitions and laws of logic.
 

FAQ: Logic of Set Operations: Proof

1. What is the logic behind set operations?

The logic of set operations involves the use of mathematical principles to manipulate sets in order to prove statements or solve problems.

2. What are the basic set operations?

The basic set operations are union, intersection, and complement. Union combines elements from two or more sets, intersection finds the common elements between two sets, and complement finds the elements that are not in a given set.

3. How do you prove set operations using Venn diagrams?

Venn diagrams are a visual representation of sets and their relationships. To prove set operations using Venn diagrams, you can shade in the areas of the diagram that represent the sets involved and show how they overlap or do not overlap.

4. Can you use algebraic equations to prove set operations?

Yes, algebraic equations can also be used to prove set operations. For example, the distributive law can be applied to set operations to show how they are related.

5. What is the importance of understanding the logic of set operations?

Understanding the logic of set operations is important because it allows us to manipulate and analyze sets in a systematic and logical way. This can be applied in various fields such as mathematics, computer science, and statistics to solve problems and make informed decisions.

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