Discrete Mathematics Answers

Write the negation in English.

a. There is a student s such that for all courses x, s like x

Translate in two ways each of these statements into logical expressions using predicates, quantifiers, and logical connectives. First, let the domain consist of the students in your class and second, let it consist of all people. a) Everyone in your class has a cellular phone. b) Somebody in your class has seen a foreign movie. c) There is a person in your class who cannot swim. d) All students in your class can solve quadratic equations. e) Some student in your class does not want to be rich

Let P(x) denote the statement “x ≤ 4.” What are these truth values?

a) P(0)

b) P(4)

c) P(6)

d) P(5)

e) P (1)

Suppose that A = {1,3,5}, B = {1,5}, C = {3,7}, and D = {1,3}. Determine which of these sets are subsets of

which other of these sets.

How many 10-bit strings contain 6 or more 1’s? Explain and show all the steps.

How many 9-bit strings (that is, bit strings of length 9) are there which:

(a) Start with the sub-string 101? Explain.

(b) Have weight 5 (i.e., contain exactly five 1’s) and start with the sub-string 101? Explain.

(c) Either start with 101 or end with 11 (or both)? Explain.

(d) Have weight 5 and either start with 101 or end with 11 (or both)?Explain

You break your piggy-bank to discover lots of pennies and nickels. You start arranging these in rows of 6 coins. Show all your steps with written explanation.

(a) You find yourself making rows containing an equal number of

pennies and nickels. For fun, you decide to lay out every possible

such row. How many coins will you need?

b) How many coins would you need to make all possible rows

of 6 coins (not necessarily with equal number of pennies and

nickels)?

Discussion Assignment

Let f(x)=\sqrt(x) with f: \mathbb{R} \to \mathbb{R}. Discuss the properties of f. Is it injective, surjective, bijective, is it a function? Why or why not? Under what conditions change this?

Explain using examples.

show that p ↔ q and (p ∧ q) ∨ (¬p ∧ ¬q) are logically equivalent

if (p ∧ q) then (q ∨ r)

¬ (p → q) ≡ p ∧ ¬q