Discrete Mathematics Answers

A home security system has a pad with 9 digit (1 to 9 ). Find the number of possible 5-digit pass code if digit can be repeated or if digit cannot be repeated

a)

i) Give an inductive formula for the sum of the first n odd numbers:

1 + 3 + 5 + ... + 2n -1

Show your induction process.

ii) Use the proof by mathematical induction to prove the correctness of your

inductive formula in i) above.

What is p ⊕ p ? true or false

Let p and q be the propositions “The total amount is discounted” and “The items have been packed,” respectively. Express each of these compound propositions as an English sentence.

Construct the truth table of (p \land \lnot q) \lor \lnot (q \land r) \lor (r \land p)

Use truth tables to prove that (p \land \lnot q) \lor \lnot (q \land r) \lor (r \land p) \iff p \lor \lnot q \lor \lnot r

Construct the truth table of (p \land \lnot q) \lor \lnot (q \land r) \lor (r \land p)

Use truth tables to prove that (p \land \lnot q) \lor \lnot (q \land r) \lor (r \land p) \iff p \lor \lnot q \lor \lnot r

Explain, without using a truth table, why (p ∨ q ∨ r) ∧

(¬p ∨ ¬q ∨ ¬r) is true when at least one of p, q, and r

is true and at least one is false, but is false when all three

variables have the same truth value.

Determine whether each of these compound propositions

is satisfiable.

a) (p ∨ ¬q) ∧ (¬p ∨ q) ∧ (¬p ∨ ¬q)

b) (p → q) ∧ (p → ¬q) ∧ (¬p → q) ∧ (¬p → ¬q)

c) (p ↔ q) ∧ (¬p ↔ q)

250 members of a certain society have voted to elect a new chairman. Each member may vote for either one or two candidates. The candidate elected is the one who polls most votes. Three candidates x, y z stood for election and when the votes were counted, it was found that: - 59 voted for y only, 37 voted for z only - 12 voted for x and y, 14 voted for x and z - 147 voted for either x or y or both x and y but not for z - 102 voted for y or z or both but not for x Required i. Present the information in a Venn diagram. (6 Marks) ii. How many voters did not vote? (4 Marks) iii. How many voters voted for x only? (2 Marks) iv. Who won the elections?

250 members of a certain society have voted to elect a new chairman. Each member may

vote for either one or two candidates. The candidate elected is the one who polls most votes.

Three candidates x, y z stood for election and when the votes were counted, it was found that:

- 59 voted for y only, 37 voted for z only

- 12 voted for x and y, 14 voted for x and z

- 147 voted for either x or y or both x and y but not for z

- 102 voted for y or z or both but not for x

Required

i. Present the information in a Venn diagram. (6 Marks)

ii. How many voters did not vote? (4 Marks)

iii. How many voters voted for x only? (2 Marks)

iv. Who won the elections? (2 Marks)