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ie was rolled twice. The first time it landed on a number that we will call
“x”. The second time it landed on a number that we will call “y”. We will symbolize some
events as follows:
A: x is odd.
B: x is a prime.
C: y is odd.
D: y is a prime.
Express the following probabilities using the notation “p(. . . )”. For example, the chance
that x and y are both odd may be expressed as p(A ∧ C).
(1) The chance that x is odd and y is even.
(2) The chance that x and y are not both odd (i.e., at least one is even).
(3) The chance that x is a prime given that it is even.
(4) The chance that x is even given that it is a prime.
(5) The chance that x and y are both primes given that they are both odd.
(6) The chance that at least one of x and y is a prime given that at least one of them is odd.
I Please prove each sequent. Each proof is worth up to 10 points.

1. A&B, B>C, C>D: A&D













2. A >(B>C) |- (A&B)>C













3. A>(B>C) |- ~C> ~(A&B)




4. ~A>B, B>D, D>A |- A














5. |- P>P
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