Use the facts that the negation of a ∀ statement is a ∃ statement and that the negation of an if-then statement is an and statement to rewrite the statement without using the words necessary or sufficient. having a large income is not a necessary condition for a person to be happy.

it`s not true that for every person, if the person is happy, the person has a large income/there are people that are happy and doesn't have large incomes

Step-by-step explanation:

let be x the group of people in the world

let be I(X)=x has a large income

let be H(x)=x is happy

Having a large income is not a necessary condition for a person to be happy.

¬∀x, H(x)-->I(x)

it`s not true that for every person, if the person is happy, the person has a large income

∃x /  H(x)^¬I(x)

there are people that are happy and doesn't have large incomes

the number of the people going.

step-by-step explanation:

the cost to buy one movie ticket is $7. if the total cost for the movie is a function of how many people go, then the input is : the number of the people going.

(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240