🚧 Started chapter 2

chapter_2/exercises.md

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+**Exercise Set 2.1**
+
+Page 74
+
+In each of 1-4 represent the common form of each argument using letters to stand
+for component sentences, and fill in the blanks so that the argument in part (b)
+has the same logical form as the argument in part (a).
+
+1.
+
+a.
+
+If all integers are rational, then the number $1$ is rational.
+
+All integers are rational.
+
+Therefore, the number $1$ is rational.
+
+b.
+
+If all algebraic expressions can be written in prefix notation ,then
+
+______.
+
+______.
+
+Therefore, $(a + 2b)(a^2 - b)$ can be written in prefix notation.
+
+2.
+
+a.
+
+If all computer programs contain errors, then this program contains an error.
+
+This program does not contain an error.
+
+Therefore, it is not the case that all computer programs contain errors.
+
+b.
+
+If ______, then ______.
+
+2 is not odd.
+
+Therefore, it is not the case that all prime numbers are odd.
+
+3.
+
+a.
+
+This number is even or this number is odd.
+
+This number is not even.
+
+Therefore, this number is odd.
+
+b.
+
+______ or logic is confusing.
+
+My mind is not shot.
+
+Therefore, ______.
+
+4.
+
+a.
+
+If the program syntax is faulty, then the computer will generate an error
+message.
+
+If the computer generates an error message, then the program will not run.
+
+Therefore, if the program syntax is faulty, then the program will not run.
+
+b.
+
+If this simple graph ______, then it is complete.
+
+If this graph ______, then any two of its vertices can be joined by a path.
+
+Therefore, if this simple graph has 4 vertices and 6 edges, then ______.
+
+5. Indicate which of the following sentences are statements.
+
+a. 1,024 is the smallest four-digit number that is a perfect square.
+
+b. She is a mathematics major.
+
+c. $128 = 2^6$
+
+d. $x = 2^6$
+
+Write the statements in 6-9 in symbolic form using the symbols $\neg$, $\wedge$,
+$\vee$ and the indicated letters to represent component statements.
+
+6. Let $s = $ "stocks are increasing" and $i = $ "interest rates are steady."
+
+a. Stocks are increasing but interest rates are steady.
+
+b. Neither are stocks increasing nor are interest rates steady.
+
+7. Juan is a math major but not a computer science major. ($m = $ "Juan is a
+   math major," $c = $ "Juan is a computer science major")
+
+8. Let $h = $ "John is healthy," $w = $ "John is wealthy," and $s = $ "John is
+   wise."
+
+a. John is healthy and wealthy but not wise.
+
+b. John is not wealthy but he is healthy and wise.
+
+c. John is neither healthy, wealthy, nor wise.
+
+d. John is neither wealthy nor wise, but he is healthy.
+
+e. John is wealthy, but he is not both healthy and wise.
+
+9. Let $p = $ "$x > 5$," $q = $ "$x = 5$," and $r = $ "$10 > x$."
+
+a. $x \geq 5$
+
+b. $10 > x > 5$
+
+c. $10 > x \geq 5$
+
+10. Let $p$ be the statement "DATAENDFLAG is off," $q$ the statement "ERROR
+    equals 0," and $r$ the statement "SUM is less than 1,000." Express the
+    following sentences in symbolic notation.
+
+a. DATAENDFLAG is off, ERROR equals 0, and SUM is less than 1,000.
+
+b. DATAENDFLAG is off but ERROR is not equal to 0.
+
+c. DATAENDFLAG is off; however, ERROR is not 0 or SUM is greater than or equal
+to 1,000.
+
+e. Either DATAENDFLAG is on or it is the case that both ERROR equals 0 and SUM
+is less than 1,000.
+
+11. In the following sentence, is the word _or_ used in its inclusive or
+    exclusive sense? A team wins the playoffs if it wins two games in a row or a
+    total of three games.
+
+Write truth tables for the statement forms 12-15.
+
+12. $\neg p \wedge q$
+
+13. $\neg (p \wedge q) \vee (p \vee q)$
+
+14. $p \wedge (q \wedge r)$
+
+15. $p \wedge (\neg q \vee r)$
+
+Determine whether the statement forms in 16-24 are logically equivalent. In each
+case, construct a truth table and include a sentence justifying your answer.
+Your sentence should show that you understand the meaning of logical
+equivalence.
+
+16. $p \vee (p \wedge q) \text{ and } p$
+
+17. $\neg (p \wedge q) \text{ and } \neg p \wedge \neg q$
+
+18. $p \vee \mathbf{t} \text{ and } \mathbf{t}$
+
+19. $p \wedge \mathbf{t} \text{ and } $
+
+20. $p \wedge \mathbf{c} \text{ and } p \vee \mathbf{c}$
+
+21. $(p \wedge q) \wedge r \text{ and } p \wedge (q \wedge r)$
+
+22. $p \wedge (q \vee r) \text{ and } (p \wedge q) \vee (p \wedge r)$
+
+23. $(p \wedge q) \vee r \text{ and } p \wedge (q \vee r)$
+
+24. $(p \vee q) \vee (p \wedge r) \text{ and } (p \vee q) \wedge r$
+
+Use De Morgan's laws to write negations for the statements in 25-30.
+
+25. Hal is a math major and Hal's sister is a computer science major.
+
+26. Sam is an orange belt and Kate is a red belt.
+
+27. The connector is loose or the machine is unplugged.
+
+28. The train is late or my watch is fast.
+
+29. This computer program has a logical error in the first ten lines or it is
+    being run with an incomplete data set.
+
+30. The dollar is at an all-time high and the stock market is at a record low.
+
+31. Let $s$ be a string of length 2 with characters from $\{0, 1, 2\}$, and
+    define statements $a$, $b$, $c$, and $d$ as follows:
+
+$a = $ "the first character of $s$ is 0"
+
+$b = $ "the first character of $s$ is 1"
+
+$c = $ "the second character of $s$ is 1"
+
+$c = $ "the second character of $s$ is 2".
+
+Describe the set of all strings for which each of the following is true.
+
+a. $(a \vee b) \wedge (c \vee d)$
+
+b. $(\neg(a \vee b)) \wedge (c \vee d)$
+
+c. $((\neg a) \vee b) \wedge (c \vee (\neg d))$
+
+Assume $x$ is a particular real number and use De Morgan's laws to write
+negations for the statements in 32-37.
+
+32. $-2 < x < 7$
+
+33. $-10 < x < 2$
+
+34. $x < 2 \text{ or } x > 5$
+
+35. $x \leq -1 \text{ or } x > 1$
+
+36. $1 > x \geq -3$
+
+37. $0 > x \geq -7$
+
+In 38 and 39, imagine that _num_orders_ and _num_instock_ are particular values,
+such as might occur during execution of a computer program. Write negations for
+the following statements.
+
+38. $(\text{num_orders } > 100 \text{ and } \text{num_instock } \leq 500) \text{ or } \text{num_instock } < 200$
+
+39. $(\text{num_orders } < 50 \text{ and } \text{num_instock } > 300) \text{ or } (50 \leq \text{ num_orders } < 75 \text{ and } \text{num_instock} > 500)$
+
+Use truth tables to establish which of the statement forms in 40-43 are
+tautologies and which are contradictions.
+
+40. $(p \wedge q) \vee (\neg p \vee (p \wedge \neg q))$
+
+41. $(p \wedge \neg q) \wedge (\neg p \vee q)$
+
+42. $((\neg p \wedge q) \wedge (q \wedge r)) \wedge \neg q$
+
+43. $(\neg p \vee q) \vee (p \wedge \neg q)$
+
+44. Recall that $a < x < b$ means that $a < x$ and $x < b$. Also $a \leq b$
+    means that $a < b$ or $a = b$. Find all real numbers that satisfy the
+    following inequalities.
+
+a. $2 < x \leq 0$
+
+b. $1 \leq x < -1$
+
+45. Determine whether the statements in (a) and (b) are logically equivalent.
+
+a. Bob is both a math and computer science major and Ann is a math major, but
+Ann is not both a math and computer science major.
+
+b. It is not the case that both Bob and Ann are both math and computer science
+majors, but it is the case that Ann is a math major and Bob is both a math and
+computer science major.
+
+46. Let the symbol $\oplus$ denote _exclusive or_; so
+    $p \plus q \equiv (p \vee q) \wedge \neg(p \wedge q)$. Hence the truth table
+    for $p \plus q$ is as follows:
+
+| $p$ | $q$ | $p \plus q$ |
+| --- | --- | ----------- |
+| T   | T   | F           |
+| T   | F   | T           |
+| F   | T   | T           |
+| F   | F   | F           |
+
+a. Find simpler statement forms that are logically equivalent to $p \plus p$ and
+$(p \oplus p) \oplus p$.
+
+b. Is $(p \oplus q) \oplus r \equiv p \oplus (q \oplus r)$? Justify your answer.
+
+c. Is $(p \oplus q) \wedge r \equiv (p \wedge r) \oplus (q \wedge r)$? Justify
+your answer.
+
+47. In logic and in standard English, a double negative is equivalent to a
+    positive. There is one fairly common English usage in which a "double
+    positive" is equivalent to a negative. What is it? Can you think of others?
+
+In 48 and 49 below, a logical equivalence is derived from Theorem 2.1.1. Supply
+a reason for each step.
+
+48.
+
+$$ p \vee \neg q \vee (p \wedge q) \equiv p \wedge (\neg q \vee q) \text{ by (a)} $$
+
+$$ \quad \equiv p \wedge (q \vee \neg q) \text{ by (b)} $$
+
+$$ \quad \equiv p \wedge \mathbf{t} \text{ by (c)} $$
+
+$$ \quad \equiv p \text{ by (d)} $$
+
+Therefore, $(p \wedge \neg q) \vee (p \wedge q) \equiv p$.
+
+49.
+
+$$ (p \vee \neg q) \wedge (\neg p \vee \neg q) $$
+
+$$ \quad \equiv (\neg q \vee p) \wedge (\neg q \vee \neg p) \text{ by (a)} $$
+
+$$ \quad \equiv \neg q \vee (p \wedge \neg p) \text{ by (b)} $$
+
+$$ \quad \equiv q \vee \mathbf{c} \text{ by (c)} $$
+
+$$ \quad \equiv \neg q \text{ by (d)} $$
+
+Therefore, $(p \vee \neg q) \wedge (\neg p \vee \neg q) \equiv \neg q$.
+
+Use Theorem 2.1.1 to verify the logical equivalences in 50-54. Supply a reason
+for each step.
+
+50. $(p \wedge \neg q) \vee p \equiv p$
+
+51. $p \wedge (\neg q \vee p) \equiv p$
+
+52. $\neg(p \vee \neg q) \vee (\neg p \wedge \neg q) \equiv \neg p$
+
+53. $\neg((\neg p \wedge q) \vee (\neg p \wedge \neg q)) \vee (p \wedge q) \equiv p$
+
+54. $(p \wedge (\neg(\neg p \vee q))) \vee (p \wedge q) \equiv p$