How do you distribute AND and OR?

Hi all,
I'm working through a book on set theory and I haven't gotten very far yet. I'm sort of struck by the looseness of the proofs for the associative, distributive, and commutative laws provided by the book and I think this stems from the fact that I'm not sure how to relate / distribute the ANDs and ORs of written logic.

Example:
If x is an element of (A intersect B) union C, then x is an element of A AND x is an element of B, OR x is an element of C.

Why does the proof jump from this kind of formulation to: (A U C) intersect (B U C)?
Why is the OR back distributed over the AND?

Any help clarifying my confusion would be much appreciated.
Thanks,
Ultros

Quote:

---

Originally Posted by Ultros88

Hi all,
I'm working through a book on set theory and I haven't gotten very far yet. I'm sort of struck by the looseness of the proofs for the associative, distributive, and commutative laws provided by the book and I think this stems from the fact that I'm not sure how to relate / distribute the ANDs and ORs of written logic.

Example:
If x is an element of (A intersect B) union C, then x is an element of A AND x is an element of B, OR x is an element of C.

Why does the proof jump from this kind of formulation to: (A U C) intersect (B U C)?
Why is the OR back distributed over the AND?

Any help clarifying my confusion would be much appreciated.
Thanks,
Ultros

---

Using for AND and for OR, we have the following properties:





We use a truth table (note that means "true" and "false"):





Since the final columns of the two truth tables are the same, we can say that the two statements are equivalent.

Using a similar method, you can show that


Thus, we can show that







(from the result above)