Discrete Mathematics Answers

1. Draw the Hasse diagram representing the partial ordering {(a, b) |a divides b} on {1, 2, 3, 4, 6, 8, 12}.

D. Let P(x) denote the statement 𝟏 𝒙 𝟐+𝟏 > 1. If its domain are all real numbers, what is the truth value of the following quantified statement? (5 pts each) 1. ∃xP(x) 2. ∀xP(x)

Let P ( x ) denote the sentence 2x + y = 5. What are the truth value of the following where the domain of x and y is the set of all integers.

1. Ɐx ⱯyP( x,y)

2. Ɐx ꓱyP ( x,y)

3. ꓱx ⱯyP ( x,y)

4. ꓱx ꓱyP( x,y)

Identify if the following statements are predicate logic. Give a domain of discourse for each propositional function. (3 items x 5 points)

The movie won the Academy Award for the past 2 years.

1 + 3 = 4

(x+2)2 is a prime number.

Solve the problem and show solutions

1.A clothing store wants to stock

sweatshirts that come in four size (small, medium,large,x-large) and in 3 colors (yellow,blue, white). How many different types of sweatshirts will the store have to stock.

2.A clothing store wants to stock sweatshirts that come in four size (small, medium,large,x-large) and in 3 colors (yellow,blue, white). How many different types of sweatshirts will the store have to stock?

3.For her birthday, Karla received a new wardrobe consisting of 5 shirts, 3 pairs of pants,2 skirts, and 3 pairs of shoes for her birthday. How many outfit can she make?

Consider the design of a communication system.

⦁ How many three-digit phone prefixes that are used to represent a particular geographic area (such as an area code) can be created from the digits 0 through 9?

⦁ As in part i., how many three-digit phone prefixes are possible that do not start with 0 or 1, but contain 0 or 1 as the middle digit?

⦁ How many three-digit phone prefixes are possible in which no digit appears more than once in each prefix?

Suppose that an operating room needs to handle three knee, four hip, and five shoulder surgeries.

⦁ How many different sequences are possible?

⦁ How many different sequences have all hip, knee, and shoulder surgeries scheduled consecutively?

⦁ How many different schedules begin and end with a knee surgery?

Number plates in an Arab country consist of 3 letters (chosen from a specified set of 20 letters from the Arabic alphabet) and 4 digits (from 0 to 9).

⦁ Find the probability of a plate having identical letters.

⦁ Find the probability of a plate having identical digits.

⦁ Find the probability of a plate having consecutive digits.

Determine whether (p ∧ (p → q)) →q is a tautology using laws of logic.

Determine whether (¬q ∧ (p → q)) → ¬p) is a tautology using laws of logic.