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1. Home Assignment II: DISCRETE MATHEMATICS
(CSC 1700-4 12 points)
Name: ____________________________________________ Matric No. _______________________
QUESTIONS
1. Use the Principle of Mathematical Induction to prove that [2 pts]
2|(𝑛𝑛2
+ 𝑛𝑛) for all 𝑛𝑛 ≥ 0.
2. Use mathematical induction to show that 𝑛𝑛 lines in the plane passing
through the same point divide the plane into 2𝑛𝑛 regions. [2 pts]
3. Suppose that 𝑄𝑄(𝑥𝑥) is the statement “𝑥𝑥 + 1 = 2𝑥𝑥. ” What are the truth values
of ∀𝑥𝑥𝑥𝑥(𝑥𝑥) and ∃𝑄𝑄(𝑥𝑥)? [2 pts]
4. Let 𝑃𝑃( 𝑚𝑚, 𝑛𝑛) be “𝑛𝑛 is greater than or equal to 𝑚𝑚” for all nonnegative
integers. What are the truth values of [2 pts]
a. ∀𝑛𝑛∀𝑚𝑚𝑚𝑚( 𝑚𝑚, 𝑛𝑛)
b. ∀𝑛𝑛∃𝑚𝑚𝑚𝑚( 𝑚𝑚, 𝑛𝑛)
c. ∃𝑛𝑛∀𝑚𝑚𝑚𝑚( 𝑚𝑚, 𝑛𝑛)
d. ∃𝑛𝑛∃𝑚𝑚𝑚𝑚( 𝑚𝑚, 𝑛𝑛)?
5. Prove or disprove that (𝑝𝑝 → 𝑞𝑞) → 𝑟𝑟 and 𝑝𝑝 → (𝑞𝑞 → 𝑟𝑟) are equivalent. [2 pts]
6. How many different license plates can be made if each license plate consists
of three letters followed by three digits or four letters followed by two digits? [2 pts]