Someone told me that the solution from cramster has someillegal moves. Can anyon

Question

Someone told me that the solution from cramster has someillegal moves. Can anyone confirm it for me? The question was: Use an element argument to prove that for all setsA,B,C,               Ax (B U C) = (A x B) U (A x C). assume that all sets are subsets of a universal set U Someone told me that the solution from cramster has someillegal moves. Can anyone confirm it for me? The question was: Use an element argument to prove that for all setsA,B,C,               Ax (B U C) = (A x B) U (A x C). assume that all sets are subsets of a universal set U

Explanation / Answer

QuestionDetails: Someone told me that the solution from cramster has someillegal moves. Can anyone confirm it for me? The question was: Use an element argument to prove that for all setsA,B,C,               Ax (B U C) = (A x B) U (A x C). assume that all sets are subsets of a universal set U
LET X BE AN ELEMENT OF LHS..A x (B U C)
HENCE
X IS AN ELEMENT OF A AND X IS AN ELEMENT OF (B U C)
THAT IS
X IS AN ELEMENT OF A AND X IS AN ELEMENT OF B ...OR.... C
CASE 1
X IS AN ELEMENT OF A AND X IS AN ELEMENT OF B
SO X IS AN ELEMENT OF (A x B)
SO X IS AN ELEMENT OF (A x B) U (A x C).
CASE 2
X IS AN ELEMENT OF A AND X IS AN ELEMENT OF C
SO X IS AN ELEMENT OF (A x C)
SO X IS AN ELEMENT OF (A x B) U (A x C).
THUS IN ALL CASES IF
X IS AN ELEMENT OF A x (B U C) , THEN IT IMPLIES
X IS AN ELEMENT OF (A x B) U (A x C).
THAT IS
A x (B U C) IS A SUB SET OF (A x B) U (A xC).............................1
SIMILARLY IF
Y IS AN ELEMENT OF (A x B) U (A x C)
CASE 1
Y IS AN ELEMENT OF A AND B
SO Y IS AN ELEMENT OF A AND (B U C)
SO Y IS AN ELEMENT OF A x (B U C)

OR....
CASE 2
Y IS AN ELEMENT OF A AND C
SO Y IS AN ELEMENT OF A AND (B U C)
SO Y IS AN ELEMENT OF A x (B U C)

THUS IN ALL CASES IF
Y IS AN ELEMENT OF (A x B) U (A x C). , THEN IT IMPLIES
X IS AN ELEMENT OF A x (B U C)
THAT IS
(A x B) U (A x C) IS A SUB SET OF A x (B U C)................2
CONCLUSIONS 1 AND 2 IMPLIES
A x (B U C) = (A x B) U (A x C).

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