Any two events $X$ and $Y$ are called mutually exclusive when the probability $P(X$ and $Y) = 0$ and they are called exhaustive when $P(X$ or $Y) = 1$. Suppose $A$ and $B$ are events and the probability of each of these two events is strictly between 0 and 1 (i.e., $0 < P(A) < 1$ and $0 < P(B) < 1$). Statements (5) $A$ and $B$ are mutually exclusive if and only if not $A$ and not $B$ are exhaustive. (6) $A$ and $B$ are independent if and only if not $A$ and not $B$ are independent. (7) $A$ and $B$ cannot be simultaneously independent and exhaustive. (8) $A$ and $B$ cannot be simultaneously mutually exclusive and exhaustive.
Any two events $X$ and $Y$ are called mutually exclusive when the probability $P(X$ and $Y) = 0$ and they are called exhaustive when $P(X$ or $Y) = 1$. Suppose $A$ and $B$ are events and the probability of each of these two events is strictly between 0 and 1 (i.e., $0 < P(A) < 1$ and $0 < P(B) < 1$).
Statements
(5) $A$ and $B$ are mutually exclusive if and only if not $A$ and not $B$ are exhaustive.\\
(6) $A$ and $B$ are independent if and only if not $A$ and not $B$ are independent.\\
(7) $A$ and $B$ cannot be simultaneously independent and exhaustive.\\
(8) $A$ and $B$ cannot be simultaneously mutually exclusive and exhaustive.