Chapter 6 Exercises

Exercise 6.1 Without changing their meanings, convert each of the following sentences into a sentence having the form “P if and only if Q.”

  1. A series converges whenever it converges absolutely.
  2. For a function to be continuous, it is sufficient that it is differentiable.
  3. For a function to be continuous, it is necessary that it is integrable.
  4. A function is rational if it is a polynomial.
  5. An integer is divisible by 8 only if it is divisible by 4.
  6. For matrix \(A\) to be invertible, it is necessary and sufficient that \(\det(A) \neq 0\).
  7. If a function has a constant derivative then it is linear, and conversely.
  8. If \(xy = 0\) then \(x = 0\) or \(y = 0\), and conversely.
  9. If \(a \in \mathbb{Q}\) then \(5a \in \mathbb{Q}\), and if \(5a \in \mathbb{Q}\) then \(a \in \mathbb{Q}\).

Exercise 6.2 Write truth tables for the following:

  1. \(p \lor (q \Rightarrow r)\)
  2. \((q \lor r) \Leftrightarrow (r \lor q)\)
  3. \(\sim(\sim p \lor q)\)
  4. \(\sim(p \lor q) \lor (\sim p)\)
  5. \(p \lor (q \land \sim r)\)

Exercise 6.3 Suppose \(p\) is false and
\((r \Rightarrow s) \Leftrightarrow (p \land q)\) is true.
Find the truth values of \(r\) and \(s\).

Exercise 6.4 Decide whether the following pairs are logically equivalent:

  1. \(p \land q\) and \(\sim(\sim p \lor \sim q)\)
  2. \((p \Rightarrow q) \lor r\) and \(\sim((p \land \sim q) \land \sim r)\)
  3. \(p \land (q \lor \sim q)\) and \((\sim p) \Rightarrow (q \land \sim q)\)
  4. \(\sim(p \Rightarrow q)\) and \(p \land \sim q\)
  5. \((p \Rightarrow q) \lor r\) and \(\sim((p \land \sim q) \land \sim r)\)

Exercise 6.5 Write down the negation of the following statements.

  1. It’s not true that the number 2 is even.
  2. The number 2 is even or the number 3 is odd.
  3. The discriminant is negative only if the quadratic equation has no real solutions.
  4. If a function has a constant derivative then it is linear, and conversely.
  5. Either you pay your tuition or you will be withdrawn from the institute.

Exercise 6.6 Let \(p, q\) and \(r\) be propositions. Let \(T\) denote a tautology.

  1. Prove the following equivalences:
  • \(p \land q \equiv q \land p\)
  • \(\neg p \land q \equiv \neg (q \land \neg p)\)
  • \(p \land T \equiv T\)
  • \(p \lor \neg p \equiv T\)
  • \(p \land (q \lor r) \equiv (p \land q) \lor (p \land r)\)
  1. Using results in part (i), show that:

\[ [(\neg r) \land p \land q] \lor [(\neg r) \land (\neg p) \land q] \equiv \neg (q \Rightarrow r) \]

Exercise 6.7 Consider the statement:
“For all integers \(n\), if \(n\) is a multiple of 6 then \(n\) is even.”

Let:

  • \(P(n)\): \(n\) is a multiple of 6
  • \(Q(n)\): \(n\) is even
  • \(\mathbb{Z}\): the set of all integers
  1. Express the statement in symbolic form.
  2. Write the converse, inverse, and contrapositive in English.
  3. Express the statement using a necessary condition.
  4. Express the statement using a sufficient condition.
  5. Express the statement using “only if”.

Exercise 6.8

  1. What is the English interpretation of the contrapositive of:

\[ \forall \epsilon > 0 \, \forall x \in D(f) \, \exists \delta > 0, \quad |f(x) - l| < \epsilon \Rightarrow |x - a| \geq \delta \]

  1. What is the English interpretation of the negation of:

\[ \exists \delta > 0 \, \forall x \in S, \quad S \cap N^c(a, \delta) = \emptyset \]

Exercise 6.9 Consider the following argument for an integer \(n\):

  • \(n\) is odd.
  • If \(n\) is not odd, then \(n^2\) is not odd.
  • Therefore, \(n^2\) is even.

Let:

  • \(p\): \(n\) is odd
  • \(q\): \(n^2\) is odd
  1. Express the argument symbolically.
  2. Determine whether the argument is valid.

Exercise 6.10 Prove each of the following statements:

    1. For all integers \(a\), \(b\), and \(c\) with \(a \neq 0\), if \(a \mid b\) and \(a \mid c\), then \(a \mid (b - c)\).
    1. For each \(n \in \mathbb{Z}\), if \(n\) is an odd integer, then \(n^3\) is an odd integer.
    1. For each integer \(a\), if \(4 \mid (a - 1)\), then \(4 \mid (a^2 - 1)\).

Exercise 6.11 Let \(n\) be an integer. Prove each of the following:

    1. If \(n\) is even, then \(n^3\) is even.
    1. If \(n^3\) is even, then \(n\) is even.
    1. The integer \(n\) is even if and only if \(n^3\) is even.
    1. The integer \(n\) is odd if and only if \(n^3\) is odd.

Exercise 6.12

    1. Write the contrapositive of the following statement:
      For all positive real numbers \(a\) and \(b\), if \(\sqrt{ab} \neq \frac{a + b}{2}\), then \(a \neq b\).
    1. Is this statement true or false?
      Prove the statement if it is true or provide a counterexample if it is false.

Exercise 6.13 Is the following proposition true or false?

For all integers \(a\) and \(b\), if \(ab\) is even, then \(a\) is even or \(b\) is even.

Justify your conclusion by writing a proof if the proposition is true or by providing a counterexample if it is false.

Exercise 6.14 Are the following propositions true or false? Justify your conclusion.

    1. There exist integers \(x\) and \(y\) such that \(4x + 6y = 2\).
    1. There exist integers \(x\) and \(y\) such that \(6x + 15y = 2\).
    1. There exist integers \(x\) and \(y\) such that \(6x + 15y = 9\).