Use properties of Boolean algebra to simplify the following Boolean expression (show-
ing all the steps):
(x + y
0
)(x
0 + y
0
)
0

Police report that 90% of drivers stopped on suspicion of drunk driving are given a
breath test, 11% are given a blood test, and 8% are given both.
(a) In this context, define two events A and B.
(b) Write the given information in probability notation.
(c) Explain, in this context, the meaning of P(A ∩ B).
(d) Are A and B disjoint events?
(e) Are A and B independent events?

In an exam, a student is required to answer 10 out of 13 questions. Find the number of
possible choices if the student must answer:
(a) the first two questions;
(b) the first or second question, but not both;
(c) exactly 3 out of the first 5 questions;
(d) at least 3 out of the first 5 questions.

(a) In how many different ways can the letters of the word wombat be arranged?
(b) In how many different ways can the letters of the word wombat be arranged if the letters wo
must remain together (in this order)?
(c) How many different 3-letter words can be formed from the letters of the word wombat? And
what if w must be the first letter of any such 3-letter word?

Decide whether or not the following quantified propositions are true or false, where
in each case the universal set is the set of positive integers N+. Justify each of your conclusions
with a proof or a counterexample.
(a) ∀n(2
n ≥ n
2
)
(b) ¬∃n(n
2 = 15)

Use mathematical induction to prove that 4 is a factor of 9n − 5
n
for all integers n ≥ 1.

(a) Use rules of inference and laws of logical equivalence to prove the following:
(p → q) ∧ (r → s) ∧ [t → ¬(q ∨ s)] ∧ t ⇒ (¬p ∧ ¬r)
(b) Use the MATLAB program truth.m and a modified version of propos.m to verify the valid-
ity of the inference in part (a).

Calculate (by hand) the appropriate truth table to prove or disprove the following:
(p → q) ∧ (¬p → r) ⇒ (q ∨ r)
If it is invalid, give a counterexample; otherwise, explain why it is valid.

Let p and q bethepropositions“Theelectionisdecided” and“Thevoteshavebeencounted,”respectively.Express each of these compound propositions as an English sentence

1.In class of 40 students, 38 offer maths, 24 offer English. Each of the students offer at least one of the two subjects. How many students offer both subjects?
2. If A={ 3 ,4 ,5 ,9 and B={1,2,6,9,7}. Find (I) A–B (ii)B n A.