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

answer: a

step-by-step explanation:

if xy ~ yz therefore xy would be the same length as yz

is only a line? i think so