Proving the Set Equation A-B-C A-B U A∩B∩C

Boolean algebra is a powerful tool for solving problems in set theory. In this article, we will use Boolean algebra to prove the equivalence of the following two statements for any sets A, B, and C:

Statements to Prove

We need to show that:

a) x in A setminus B setminus C b) x in A setminus B cup A intersect B intersect C

Reformulating the Statements as Propositional Formulas

To begin our proof, we will reformulate the above statements as propositional formulas by using the definitions of set operators.

Statement a)

The statement a) can be reformulated as:

a.i) x in A wedge x not in B setminus C a.ii) x in A wedge neg x in B setminus C a.iii) x in A wedge neg x in B wedge x not in C

Statement b)

The statement b) can be reformulated as:

b.i) x in A setminus B vee x in A intersect B intersect C b.ii) x in A wedge x not in B vee x in A intersect B intersect C b.iii) x in A wedge x not in B vee x in A wedge x in B wedge x in C

Provability and Set Operators

At this point, we have not yet proven the equivalence between the two statements. We have only reformulated them. Our task now is to provide a proof for the equivalence of the following two statements:

a.iii) x in A wedge neg x in B wedge x not in C b.iii) x in A wedge x not in B vee x in A wedge x in B wedge x in C

The proof can be done with relative ease. Let's proceed with the proof:

a.iii) x in A wedge neg x in B wedge x not in C a.iii) x in A wedge neg x in B wedge neg x in C a.iii) x in A wedge neg x in B wedge neg neg x in C a.iii) x in A wedge neg x in B wedge x in C a.iii) x in A wedge (x in C vee x in C) a.iii) x in A wedge (x in C vee true) a.iii) x in A wedge (true vee x in C) a.iii) x in A wedge (true vee x in C vee x in B wedge neg x in B) a.iii) x in A wedge (true vee x in C vee x in B wedge x in B wedge neg x in B) a.iii) x in A wedge (x not in B vee x in C vee x in C) a.iii) x in A wedge (x not in B vee x in B wedge x in C) a.iii) x in A wedge (x not in B vee x in B wedge x in C)

Now, we need to show that this is equivalent to b.iii). We follow a similar process:

b.iii) x in A wedge x not in B vee x in A wedge x in B wedge x in C b.iii) x in A wedge (x not in B vee x in B wedge x in C) b.iii) x in A wedge (x not in B vee x in B wedge x in C)

Thus, we have shown that the two statements are equivalent.

Visualizing with Venn Diagram

We can use a Venn Diagram to visualize the sets involved. A Venn Diagram consists of:

8 pieces representing all possible cases: A AB' C' U AB' C U ABC' U ABC B A' BC' U ABC' U A' BC U ABC C A' B' C U AB' C U A' BC U ABC B - C A' BC' U ABC' A - B - C AB' C' U AB' C U ABC A - B AB' C' U AB' C A - B U ABC AB' C' U AB' C U ABC

From the Venn Diagram, it is clear that:

A - B - C AB' C' U AB' C U ABC A - B U ABC AB' C' U AB' C U ABC

Hence, both sets are equal.

Using Boolean Logic

We can also use Boolean Logic to prove the same. Let's represent:

a AND not b AND not c ab' c' a AND not b OR a AND b AND c ab' abc ab' a(b' OR c) ab' ac ac ac(true OR c) acb acb' abc ab'c ab'c' ab'c abc ab' abc

This confirms that the left-hand side and right-hand side are equivalent in Boolean Logic as well.

Concluding Thoughts

Boolean algebra and Venn diagrams are two powerful tools for understanding and proving set equations. By breaking down the problem into manageable parts, we can solve what initially looked like a complex problem with relative ease. As illustrated, the logical and visual methods offer a clear and effective way to solve such problems, making them invaluable in set theory and beyond.