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2.7. Let P(n): n and n + 2 are primes. be an open sentence over the domain N. Find six positive integers n for which P(n) is true. If n ? N such that P(n) is true, then the two integers n, n + 2 are called twin primes. It has been conjectured that there are infinitely many twin primes. 2.8. Let P(n): (n²+5n+6)/2 is even. (a) Find a set S? of three integers such that P(n) is an open sentence over the domain S? and P(n) is true for each n ? S?. (b) Find a set S? of three integers such that P(n) is an open sentence over the domain S? and P(n) is false for each n ? S?. 2.9. Find an open sentence P(n) over the domain S = {3, 5, 7, 9} such that P(n) is true for half of the integers in S and false for the other half. 2.10. Find two open sentences P(n) and Q(n), both over the domain S = {2, 4, 6, 8}, such that P(2) and Q(2) are both true, P(4) and Q(4) are both false, P(6) is true and Q(6) is false, while P(8) is false and Q(8) is true. Section 2.2: The Negation of a Statement 2.11. State the negation of each of the following statements. (a) ?2 is a rational number. (b) 0 is not a negative integer. (c) 111 is a prime number. 2.12. Complete the truth table in Figure 2.16. 2.13. State the negation of each of the following statements. (a) The real number r is at most ?2. (b) The absolute value of the real number a is less than 3. (c) Two angles of the triangle are 45°.

          2.7. Let P(n): n and n + 2 are primes. be an open sentence over the domain N. Find six positive integers n for which P(n) is true. If n ? N such that P(n) is true, then the two integers n, n + 2 are called twin primes. It has been conjectured that there are infinitely many twin primes.
2.8. Let P(n): (n²+5n+6)/2 is even.
(a) Find a set S? of three integers such that P(n) is an open sentence over the domain S? and P(n) is true for each n ? S?.
(b) Find a set S? of three integers such that P(n) is an open sentence over the domain S? and P(n) is false for each n ? S?.
2.9. Find an open sentence P(n) over the domain S = {3, 5, 7, 9} such that P(n) is true for half of the integers in S and false for the other half.
2.10. Find two open sentences P(n) and Q(n), both over the domain S = {2, 4, 6, 8}, such that P(2) and Q(2) are both true, P(4) and Q(4) are both false, P(6) is true and Q(6) is false, while P(8) is false and Q(8) is true.

Section 2.2: The Negation of a Statement
2.11. State the negation of each of the following statements.
(a) ?2 is a rational number.
(b) 0 is not a negative integer.
(c) 111 is a prime number.
2.12. Complete the truth table in Figure 2.16.
2.13. State the negation of each of the following statements.
(a) The real number r is at most ?2.
(b) The absolute value of the real number a is less than 3.
(c) Two angles of the triangle are 45°.

2.7. Let P(n): n and n + 2 are primes. be an open sentence over the domain N. Find six positive integers n for which P(n) is true. If n ? N such that P(n) is true, then the two integers n, n + 2 are called twin primes. It has been conjectured that there are infinitely many twin primes.
2.8. Let P(n): (n²+5n+6)/2 is even.
(a) Find a set S? of three integers such that P(n) is an open sentence over the domain S? and P(n) is true for each n ? S?.
(b) Find a set S? of three integers such that P(n) is an open sentence over the domain S? and P(n) is false for each n ? S?.
2.9. Find an open sentence P(n) over the domain S = 3, 5, 7, 9 such that P(n) is true for half of the integers in S and false for the other half.
2.10. Find two open sentences P(n) and Q(n), both over the domain S = 2, 4, 6, 8, such that P(2) and Q(2) are both true, P(4) and Q(4) are both false, P(6) is true and Q(6) is false, while P(8) is false and Q(8) is true.

Section 2.2: The Negation of a Statement
2.11. State the negation of each of the following statements.
(a) ?2 is a rational number.
(b) 0 is not a negative integer.
(c) 111 is a prime number.
2.12. Complete the truth table in Figure 2.16.
2.13. State the negation of each of the following statements.
(a) The real number r is at most ?2.
(b) The absolute value of the real number a is less than 3.
(c) Two angles of the triangle are 45°.

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