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)
please help with these proofs im not sure how the form of proofs about cartesian products should be
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
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