CS 225: Midterm Exam Solutions
Write the negation of the following three statements:
(a) If x > 0 and x2 < 0, then x ≤ −1.
(b) Having a mind is necessary for being a person.
(c) n is prime, and n is neither odd nor e
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CS 225: Midterm Exam Solutions
Write the negation of the following three statements:
(a) If x > 0 and x2 < 0, then x ≤ −1.
(b) Having a mind is necessary for being a person.
(c) n is prime, and n is neither odd nor equal to 2.
2. Let P, Q, and R be the propositions
P: "you get an A on the final examination."
Q: "you do every exercise in the book."
R: "you get an A in the class."
Translate the following English sentences into compound logical propositions.
a) Doing every exercise in the book and getting an A on the final examination are sufficient and
necessary conditions to get an A in the class.
b) You will get an A in the final examination only if you have done every exercise in the book.
Question 2:
Let B(x), W(x), and S(x) be the predicates
B(x) : "x is a good basketball player."
W(x) : "x is a good weight lifter."
S(x) : "x is strong."
Express each of the following English sentences in terms of B(x), W(x), S(x), quantifiers, and
logical connectives. Assume the domain is all people.
(You may need to use these symbols: ≥ ≤ ≠ ¬ ∧ ∨ ⊕ ≡ → ↔ ∃ ∀ )
a) Not all good weight lifters are strong.
b) Some good weight lifters are not good basketball players.
c) If someone is not strong then he is neither a good basketball player nor a good weight lifter.
d) Every person who is a good basketball player is also a good weight lifter.
Question 3:
Use truth table method to prove or disprove that the two compound propositions (P ( Q R))
and ((P ¬R) Q) are logically equivalent.
Question 4:
Use a proof by contraposition to show that if n - 5m is odd, then n and m are of opposite
parity. (Hint: Parity indicates whether a number is odd or even. Use both cases.) Question 5:
Prove by contradiction that the sum of a rational number and an irrational number is irrational. Question 6:
Based on the diagram, indicate which of the following relationships are true and which are false .
Question 8:
Construct an algebraic proof that for all sets A and B,
((Ac ∪ Bc) − A)c = A
Cite a property from Theorem 6.2.2 for every step of the proof. Theorem 6.2.2.png
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