Chapter 6 Quiz 1. � What is the cardinality of the empty set? (a) 0 (b) 1 (c) Infinity (d) It is not defined. 1. � An equivalence relation is (a) symmetric. (b) distributive. (c) denumerable. (d) All of the above 1. � Look back at Problems 6-1 and 6-2, and their solutions. Which rule of propositional logic can be applied directly to Solution 6-1 (treating Solution 6-1 as a theorem), in order to solve Problem 6-2 by the second method mentioned? Look back at Chapter 1 if you must. (a) DeMorgan’s law for implication. (b) The law of implication reversal. (c) The commutative law for disjunction. (d) The commutative law for conjunction. 1. � At the end of Solution 6-3 and in the next-to-last line of Table 6-3, you are told that you’ll get a chance to provide the reason for the statement. Now is the time! What is the reason? (a) The element u has not been specified, so we cannot say it is in the intersection of two specific sets. (b) We are making a generalization about the element u, so we cannot say anything specific about it. (c) No matter what u happens to be, it can’t be in a set that has no elements. (d) Sets S and T are not the same, so obviously element u cannot be in both of them. 1. � Refer to Solution 6-5 and Tables 6-5A and 6-5B. In the first part of the proof and in Table 6-5A, one of the steps makes the claim that r ∈ Q ∪ X. In the second part of the proof and in Table 6-5B, one of the steps says that s ∈ S ∪ Y. The reason in both cases is the same, but it’s left out, and you’re told you’ll get a chance to identify it. Your chance has come! What is the reason for this statement in both parts of the proof? (a) It follows from the definition of logical disjunction. (b) It follows from the definition of logical implication. (c) It follows from the definition of set intersection. (d) It follows from the definition of set union. 1. � Refer to Solution 6-6 and Table 6-6. In this proof, one of the statements says that h is an integer (that is, h ∈ Z). What allows us to make this claim? (a) The product-of-integers axiom. (b) The product-of-fractions axiom. (c) The sum-of-integers axiom. (d) The sum-of-fractions axiom. 1. � Refer to Solution 6-7 and Table 6-7. One of the statements says that −c = −1 × c. What allows us to make this claim? (a) The additive-inverse axiom. (b) The sum-of-integers axiom. (c) The product-of-integers axiom. (d) The definition of a rational number. 1. � Refer to Solution 6-8 and Table 6-8. One of the statements says that (ac)/(bd) ∈ Q. A similar situation occurs in Solution 6-9 and Table 6-9, where one of the statements says that (ad)/(bc) ∈ Q. What allows us to make these claims? (a) The additive-inverse axiom. (b) The sum-of-integers axiom. (c) The product-of-integers axiom. (d) The definition of a rational number. 1. � Refer to Solution 6-10 and Table 6-10. One of the statements says the following: (r + s)t + (r + s)u = t(r + s) + u(r + s) After this statement, you’re told you’ll get a chance to provide the reason. Your chance has come! What allows us to make this claim? (a) The commutative axiom for addition, applied twice. (b) The commutative axiom for multiplication, applied twice. (c) The equality axiom. (d) The distributive axiom. 1. � Refer again to Solution 6-10 and Table 6-10. In the last step, it says that a certain component of the equality axiom is used repeatedly to come to the conclusion shown. You’re told you’ll get a chance to identify that component. Your chance has come! Which component is it? (a) The reflexive property. (b) The symmetric property. (c) The transitive property. (d) The equivalence property.
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