Unit 7 Exercises

1. Let Q(x, y) denote the statement "x is greater than y."  What are the truth values of the following?

  1. Q (3, 1)
  2. Q (5, 5)
  3. Q (6, -6)
  4. Q (28, 256)

2. Let P(x) be the statement "x is happy," where the universe of discourse for x is the set of students.  Express each of the following quantifications in English.

  1. exists.gif (61 bytes) x P(x)
  2.  all.gif (70 bytes) x not.gif (54 bytes)P(x)
  3. exists.gif (61 bytes) x not.gif (54 bytes) P(x)
  4. not.gif (54 bytes) all.gif (70 bytes) x not.gif (54 bytes) P(x)

3. Let P(x) be the statement " x > x2."  If the universe of discourse is the set of real numbers, what are the truth values of the following?

  1. P(0)
  2. P(1/2)
  3. P(2)
  4. P(-1)
  5. exists.gif (61 bytes) x P(x)
  6. all.gif (70 bytes) x P(x)

4. Suppose that the universe of discourse of the atomic formula P(x,y) is {1, 2, 3}.   Write out the following propositions using disjunctions andconjunctions.

  1. all.gif (61 bytes) x P(x, 2)
  2. exists.gif (70 bytes) y P(3, y)
  3. all.gif (70 bytes) x all.gif (70 bytes) y P(x, y)
  4. exists.gif (61 bytes) xexists.gif (61 bytes) y P(x, y)
  5. exists.gif (61 bytes) x all.gif (70 bytes) y P(x, y)
  6. all.gif (70 bytes) y exists.gif (61 bytes) x P(x, y)

 

Answers for these exercises