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## Write the contrapositive of that statement

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Discrete Mathematics Assignment Questions –
Q1) State whether each of the following is a statement or is not a statement and explain why. If it is a statement, give its truth value.
a) Drink more water.
b) Paris is the capital city of the United States of America.
c) Is it going to rain tomorrow?
Q2) Consider the two propositions.
p: We can buy a book.
q: We can go to a restaurant.
Write each of the following statements in symbolic notation and as English sentences.
a) The conjunction (∧) of p and q.
b) The disjunction (∨) of p and q.
c) The negation (~) of the conjunction (∧) of p and q.
d) The negation (~) of the disjunction (∨) of p and q.
Q3) Write the statement “Every number is more than its reciprocal” symbolically by first defining a predicate and then using a quantifier.
Q4) Let P(n): n2 = n + 6.
a) What is P(2) as a statement?
b) What is P(3) as a statement?
c) What is the truth value of ∀n P(n)?
d) What is the truth value of ∃n P(n)?
Q5) Complete a truth table for (p ∧ ~q) ∨ (~p ∧ q). There are multiple ways to set up the columns of a truth table, so you may need fewer or more columns than shown.
Q6) Use the following:
p: I will watch TV.
q: I have finished my homework.
Write each of the following statements in terms of p, q, and logical connectives.
a) I will watch TV if I have finished my homework.
b) I will watch TV only if I have not finished my homework.
c) I will watch TV is a necessary condition for I have finished my homework.
d) I will not watch TV is a sufficient condition for I have finished my homework.
e) I will watch TV if and only if I have finished my homework.
Q7) Consider the following statement: If it is Friday, then Emily will go to the museum.
a) Write the contrapositive of that statement.
b) Write the converse of that statement.
Q8) Construct a truth table for (p ∧ q) ⇒ (p ∨ q). Explain how this truth table shows whether this statement is a tautology, a contradiction (absurdity), or a contingency.
Q9) Write each of the arguments below symbolically and then explain whether it is valid or not.
a) If it is hot outside, then I will go swimming.
I will not go swimming.
Therefore It is not hot outside.
b) If it is not hot outside or if it is raining, then I will not go swimming.
It is not raining.
Therefore I will not go swimming.
c) I will go swimming if and only if it is hot outside.
I will not go swimming.
Therefore It is not hot outside.
10) Prove or disprove that if the product of two numbers (in N) is even, then at least one of them must be even.
Q11) Prove or disprove that if the sum of two numbers (in N) is even, then at least one of them must be even.

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