CSC28 Fall 2020 Prepositional Logic
(total 50 points)
Dr Jagan Chidella
Please answer the following questions.
1) (i) Let p denote “He is rich” and let q denote “He is happy.” Write each statement in symbolic form using p and q. Note that “He is poor” and “He is unhappy” are equivalent to ¬p and ¬q, respectively.
(a) If he is rich, then he is unhappy.
(b) He is
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CSC28 Fall 2020 Prepositional Logic
(total 50 points)
Dr Jagan Chidella
Please answer the following questions.
1) (i) Let p denote “He is rich” and let q denote “He is happy.” Write each statement in symbolic form using p and q. Note that “He is poor” and “He is unhappy” are equivalent to ¬p and ¬q, respectively.
(a) If he is rich, then he is unhappy.
(b) He is neither rich nor happy.
(c) It is necessary to be poor in order to be happy.
(d) To be poor is to be unhappy.
(ii) Let p be “It is cold” and let q be “It is raining”. Give a simple verbal sentence which describes each of the following statements:
(a) ¬p
(b) p ∧ q
(c) p ∨ q
(d) q ∨ ¬p. In each case, translate ∧, ∨, and ∼ to read “and,” “or,” and “It is false that” or “not,” respectively, and then simplify the English sentence.
(4 + 4) points + 2 points for neatness.
2) Construct the truth table of:
(2 points) + 1 point for neatness.
3) (a) Verify that the proposition (p ∨ ¬(p ∧ q) V q) and proposition ((¬p↔¬q) ↔ (q↔r)) is a tautology or not. (using truth tables)
(b) Consider the conditional proposition p → q. The simple propositions q → p, ¬p → ¬q and ¬q → ¬p are called, respectively, the converse, inverse, and contrapositive of the conditional p → q. Find if any of these propositions are logically equivalent to p → q? (using truth tables)
(2 + 2 points) + 2 points for neatness.
4) Give the inverse, contrapositive, and converse for each of the following statements:
(a) If it rained last night, then the sidewalk is wet.
(b) If I ride a roller coaster then I will get sick.
(c) If 2 is a prime number, then 7 is an even number.
(d) If two angles are congruent, then they have the same measure.
(e) If 8 < 6, then 6 < 4.
(5 points + 1 point for neatness)
5) Translate each English sentence into a logical expression using the propositional variables defined below. Then negate the entire logical expression using parentheses and the negation operation. Apply De Morgan's law to the resulting expression and translate the final logical expression back into English.
R: the candidate has written permission from his guardian.
S: the candidate is at least 20 years old
T: the candidate is at least 14 years old
(a) The candidate has written permission from his guardian and is at least 14 years old.
(b) The candidate has written permission from his guardian or is at least 20 years old.
(2 points) + 1 point for neatness
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