Due Monday, 10/16:
- Let and (the set of all ordered pairs of 0s and 1s). Find a bijection between A and B.
- Let A, B, and C be sets. if and are functions, the composition is defined as . Show the following:
- If and are injections, then is an injection.
- If and are surjections, then is a surjection.
- If is a bijection, then defined by is a bijection (g is called the inverse of f).
- Suppose our universe is , P(x, y) is x < y (note it’s not less than or equal to), and Q(x, y, z) is x + y = z. Determine if the following statements are true or false, and explain why:
- Find the negation of the following statement: . (Hint: use the law of double negation: .) Determine which statement (the original or its negation) is true if the universe is and P(x, y) is “x – y is an even integer”.
Suggested (practice) problems (do not turn these in, but you should do them for practice):
Section 1.4 (page 53) #1-3, 8, 11-16, 50, 51
Section 1.5 (page 64) #1-2, 9, 26-32, 39-42
Challenge: Suppose our universe is and P(x, y) is x < y. Write a statement that is true in this universe that is not true in the universe . (Hint: in the integers, there is no number between 0 and 1. But this is not true in ).