Answer
a)We proved that the outcomes in \[A\cup (B\cap C)\] are the same as the outcomes in \[(A\cup B)\cap (A\cup C)\].
b)We proved that the outcomes in \[A\cap (B\cup C)\] are the same as the outcomes in \[(A\cap B)\cup (A\cap C)\].
Work Step by Step
(a)
Let A, B and C be any three events defined on a sample space S.
If ‘m’ is a member or outcome of \[A\cup (B\cap C)\] then m belongs to A or to\[(B\cap C)\] If it is a member of A or of \[(B\cap C)\] then it belongs to \[(A\cup B)\]and to \[(A\cup C)\] Thus, it is a member of \[(A\cup B)\cap (A\cup C)\].
Conversely, choose ‘m’ in\[(A\cup B)\cap (A\cup C)\]. If it belongs to A then it belongs to\[A\cup (B\cap C)\]. If it does not belong to A, then it must be a member of \[(B\cap C)\] In that case it also is a member of \[A\cup (B\cap C)\].
Therefore, we proved that the outcomes in \[A\cup (B\cap C)\] are the same as the outcomes in \[(A\cup B)\cap (A\cup C)\].
(b)
Let A, B and C be any three events defined on a sample space S.
If ‘m’ is a member or outcome of \[A\cap (B\cup C)\], then m belongs to A and to\[(B\cap C)\]. If it is a member of A and of \[(B\cap C)\], then it belongs to \[(A\cap B)\]or to \[(A\cap C)\]. Thus, it is a member of \[(A\cap B)\cup (A\cap C)\].
Conversely, choose ‘m’ in\[(A\cap B)\cup (A\cap C)\]. If it belongs to \[(A\cap B)\]or to \[(A\cap C)\]. Thus it also is a member of \[A\cap (B\cup C)\].
Therefore, we proved that the outcomes in \[A\cap (B\cup C)\] are the same as the outcomes in \[(A\cap B)\cup (A\cap C)\].
