1. ## cartesian product proofs

A X (B U C)= (A X B) U (A X C)

proof:
Assume (x,y) is an element of A X (B U C). This means x is an element of A and y is an element of B or y is an element of C. Since (x,y) can be x as an element of A and y as an element of B, (x,y) is an element of A X B. Since (x,y) can also have x as an element of A and y as an element of C, (x,y) is an element of A X C.

A X (B-C)= (A X B) - (A X C)
not sure

A X (B intersect C)= (A X B) intersect (A X C)

A X (B-C)= (A X B) - (A X C)
not sure
Here is one. They all work about the same way.
$\displaystyle \begin{gathered} (x,y) \in A \times (B\backslash C) \hfill \\ x \in A \wedge y \in B \wedge y \notin C \hfill \\ \left( {x \in A \wedge y \in B} \right) \wedge \left( {x \in A \wedge y \notin C} \right) \hfill \\ (x,y) \in A \times B \wedge (x,y) \notin A \times C \hfill \\ (x,y) \in \left( {A \times B} \right)\backslash \left( {A \times C} \right) \hfill \\ \end{gathered}$

3. is a slash equal to the difference
also is the first 1right

4. i dont understand

i dont understand
Maybe you should try to get some one-on-one help from your instructor.

6. A X (B U C)= (A X B) U (A X C)
Proof:
(x,y) E A X (B U C).
x E A and y E (B U C)
X E A and (y E B U y E C)
(x E A and y E B) or (x E A and y E C)
(x,y) E A X B or (x,y) E A X C
(x,y) E (A X B) or (A X C)

and the argument can be reversed

the only thing I cant justify is how I go from step 3 to step 4 and how I go from step 4 to step 3.

E= element of

7. ## cartesiian proof

Proof:
(x,y) E A X (B intersect C)
x E A and y E (B intersect C)
x E A and y E B and y E C
(x E A and y E B) and (x E A and y E C)
(x,y) E A X B and (x,y) E A X C
(x,y) E (A X B) intersect (A X C)

again i am only not sure how to justify going from step 3 to step 4 and how to go from step 4 to step 3

the argument reverses to prove they are equal

E= element of

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# real analysis prove that Ax(BuC)=(AxB)u(AxC)

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