离散习题讲解（1）

她来听我的演唱会......

Q1: Which of these are propositions? What are the truth values of those that are propositions?

a) If 2+2=6, then 3 is an odd number.

  • Is it a proposition? Yes.
  • Truth value? True.
    • Explanation: This is a conditional statement. The premise "2+2=6" is false, but a conditional statement where the premise is false is always true (because an implication with a false antecedent is considered true regardless of the consequent).

b) If 1+1=2, the sun will rise in the west.

  • Is it a proposition? Yes.
  • Truth value? False.
    • Explanation: This is also a conditional statement. The premise "1+1=2" is true, but the conclusion "the sun will rise in the west" is false, so the whole statement is false.

c) 4+x=5.

  • Is it a proposition? No.
  • Explanation: This is not a proposition because it contains a variable \(x\), so its truth value depends on the value of \(x\). A proposition must have a definite truth value (true or false), and this expression doesn't.

d) Boston is the capital of Massachusetts.

  • Is it a proposition? Yes.
  • Truth value? True.
    • Explanation: This is a declarative statement that can be evaluated as true or false. It is true because Boston is indeed the capital of Massachusetts.

e) Answer this question.

  • Is it a proposition? No.
  • Explanation: This is an imperative sentence, not a statement that can be true or false. Propositions must be declarative and have a truth value.

Summary of Answers:

    1. Yes, True
    1. Yes, False
    1. No
    1. Yes, True
    1. No

Q2: Let $ p $, $ q $, and $ r $ be the propositions:

  • $ p $: You get an A on the final exam.
  • $ q $: You do every exercise in this book.
  • $ r $: You get an A in this class.

Write these propositions using $ p $, $ q $, and $ r $ and logical connectives (including negations).

a) You get an A in this class, but you do not do every exercise in this book.

  • Answer: $ r q $
  • Explanation: The phrase "You get an A in this class" translates to $ r $, and "you do not do every exercise in this book" translates to $ q $. The word "but" indicates a conjunction (AND), so we use $ r q $.

b) You get an A on the final, you do every exercise in this book, and you get an A in this class.

  • Answer: $ p q r $
  • Explanation: This is simply a conjunction of three statements: "You get an A on the final" ($ p \(), "you do every exercise" (\) q \(), and "you get an A in this class" (\) r $).

c) To get an A in this class, it is necessary for you to get an A on the final.

  • Answer: $ r p $
  • Explanation: The phrase "it is necessary for you to get an A on the final" implies that getting an A on the final is a condition for getting an A in the class. This is a conditional statement: "If you get an A in this class" ($ r \(), "then you must get an A on the final" (\) p $).

d) You get an A on the final, but you don’t do every exercise in this book; nevertheless, you get an A in this class.

  • Answer: $ p q r $
  • Explanation: The word "but" indicates a conjunction. "You get an A on the final" is $ p $, "you don’t do every exercise" is $ q $, and "you get an A in this class" is $ r $. So, the complete expression is $ p q r $.

e) Getting an A on the final and doing every exercise in this book is sufficient for getting an A in this class.

  • Answer: $ (p q) r $
  • Explanation: "Sufficient for" means that if you meet the conditions on the left (getting an A on the final and doing every exercise), it guarantees the outcome on the right (getting an A in the class). This is a conditional statement: "If you get an A on the final and do every exercise" ($ p q \(), "then you will get an A in the class" (\) r $).

f) You will get an A in this class if and only if you either do every exercise in this book or you get an A on the final.

  • Answer: $ r (q p) $
  • Explanation: "If and only if" means a biconditional relationship. This says that you will get an A in the class ($ r \() if and only if you either do every exercise (\) q \() or you get an A on the final (\) p $). This is represented as $ r (q p) $.

g) If you don’t do every exercise in this book, you will not get an A in this class.

  • Answer: $ q r $
  • Explanation: This is a conditional statement. "If you don’t do every exercise" is $ q $, and "you will not get an A in this class" is $ r $. So, the statement is $ q r $.

h) You get an A on the final exam when you get an A in this class.

  • Answer: $ r p $
  • Explanation: "When" suggests a causal or conditional relationship. This can be interpreted as: "If you get an A in the class" ($ r \(), "then you get an A on the final" (\) p $), so the expression is $ r p $.

Summary of Answers:

    1. $ r q $
    1. $ p q r $
    1. $ r p $
    1. $ p q r $
    1. $ (p q) r $
    1. $ r (q p) $
    1. $ q r $
    1. $ r p $

Problem:

Simplify the following logical expressions and explain your steps:

  1. $ (p (q r)) $
  2. $ ((p) (q)) (p r) $
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