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4. 5. 6. Prove that p (gp) is a tautology using the laws of propositional logic. Prove that p((pVq) → q) is a tautology using the laws of propositional logic. Let us say a natural number n is ok if there are two natural numbers whose sum is n and whose product is n. (Convention: the natural numbers consist of 0, 1, 2,...) (a) Give a logical expression that means "n is ok". (b) Show that 0 and 4 are both ok. (c) Give a logical expression that means "every natural number is ok". (d) Give a logical expression that means "it is not the case that every number is ok". Push the negations into the expression as far as possible.

7. Let E(x, y) be a two-variable predicate meaning "x likes to eat y", where the domain of x is people and the domain of y is foods. Write logical expressions that represent the following English propositions: (a) Alice doesn't like to eat pizza. (b) Everybody likes to eat at least one food. (c) Every student likes to eat at least one food other than pizza. (d) Everyone other than Alice likes to eat at least two different foods. (e) There are two different people that like to eat the same food.

Show all steps. Correct answer is 1/2sec(theta) +Ccos(theta)

Main/84589a48-b934-4b6e-a6e1-a5d75f559df9?transferToken=MxNewDS NGIFSFh5aZ5MQMzuGzB.. Critics review films out of 5 based on three attributes: the story, the special effects and the acting. The ratings of four critics for a film are collected in the table below. B Special Critic Critic 1 Story rating Effects rating Acting rating 4.4 4.3 4.5 Critic 2 4.1 4.2 3.9 Critic 3 3.9 4 3.4 Critic 4 4.2 4.1 4.2 44° X Critic 1 also gave the film a rating for the Director's ability. If the average of Critic 1's ratings was 4.3 what rating did they give to the Director's ability? 3.9 4.0 OOOO Esc 4.1 4.2 Q Search L 2 A AR F2 +4 % x Ö- 1+ FS F4 F5 F6 F7 F8 £ %5 14 96 T 27 F9 화요 4 * 8 F10 U F11 61 UTROGENA apid Cle 2-in-1 & Fade Toner TOCLEAR FADES The look of ENG 17:51 UK 25/01/2025 TYLITTLETHING DY MIST ONIKI ALM # 공☆ Home End F12 Insert Delete +11 K P

eminder to complete the Case X T Civil Service Numerical Test Sec X C T Casework Skills Practice Test x + euauthoring.panpowered.com/DeliveryWeb/Civil Service Main/84589a48-b934-4b6e-a6e1-a5d75f559df9?transferToken=MxNewOS NGIFSFH5aZSMQMzuGzB The table shows the value of the FTSE 100 share index over 6-month periods for 2 years. Month Value (£) January 2017 7,147 July 2017 7,455 January 2018 7,692 July 2018 7,668 January 2019 7,154 If the value of shares held in the FTSE 100 was £2,450 in July 2017, what would it be in July 2018? £2,478 £2,505 O O O O C £2,520 £2,553 £2,560 EN Q Search பல் L -☀- F5 0+ F6 C F8 27 8 화요 * 0 F10 E 61 F12

(c) Suppose that all the pages are black and white, but each group of 25 con- secutive pages (1-25, 26-50, 51-75, 76-100, 101-125) must be assigned to the same printer. Each printer can be assigned 0, 25, 50, 75, 100, or 125 pages to print. How many ways are there for the 125 pages to be assigned to the five printers?

Ruff, Inc. makes dog food out of chicken and grain. Chicken has 10 grams of protein and 5 grams of fat per ounce, and grain has 2 grams of protein and 2 grams of fat per ounce. A bag of dog food must contain at least 222 grams of protein and at least 162 grams of fat. If chicken costs 11¢ per ounce and grain costs 1¢ per ounce, how many ounces of each should Ruff use in each bag of dog food to minimize cost? (If an answer does not exist, enter DNE.)

Solve the linear system of equations attached using Gaussian elimination (not Gauss-Jordan) and back subsitution. Remember that: A matrix is in row echelon form if Any row that consists only of zeros is at the bottom of the matrix. The first non-zero entry in each other row is 1. This entry is called aleading 1. The leading 1 of each row, after the first row, lies to the right of the leading 1 of the previous row.