Let p and q be the propositions. p : I bought a lottery ticket this week. q : I won the million dollar jackpot. Express each of these propositions as an English sentence. a) ¬p b) p ∨ q c) p → q d) p ∧ q e) p ↔ q f ) ¬p → ¬q g) ¬p ∧ ¬q h) ¬p ∨ (p ∧ q)

a)  ¬p : I didn't buy a lottery ticket this week.

b) p ∨ q: I bought a lottery ticket this week or I won the million dollar jackpot.

c) p → q: If I didn't buy a lottery ticket this week, then I won the million dollar jackpot.

d) p ∧ q:  I bought a lottery ticket this week and I won the million dollar jackpot.

e) p ↔ q: I bought a lottery ticket this week if and only if I won the million dollar jackpot.

f ) ¬p → ¬q: If I didn't buy a lottery ticket this week, then I didn't win the million dollar jackpot.

g) ¬p ∧ ¬q: I didn't buy a lottery ticket this week and I didn't win the million dollar jackpot.

h) ¬p ∨ (p ∧ q): I didn't buy a lottery ticket this week or I bought a lottery ticket this week and I won the million dollar jackpot.

Step-by-step explanation:

In logic, a word or group of words that joins two or more propositions together to form a connective proposition it's called connective, also called sentential connective, or propositional connective

1. Negation: the symbol is ¬, or ~. It is use for saying that the proposition is false. This connective proposition only affects one statement.

2. Disjunction ("or"): the symbol is ∨. It is use for saying that at least one of the propositions are true.

3. Conjunction ("and"): the symbol is ∧. It is use for saying that both of the propositions, at the same time are true.

4. Conditional (“if . . . then”): the symbor is →. In this structure the first proposition it's called antecedent and the second one consecuent. For this connective the only case when it's not true is when the antecendent is true and the consecuent is false.

5. Biconditional ("if and only if"): the symbol is  ↔. This structure is a double conditional. And the proposition is true when antecent and consecuent are both true or both false.

easy its decreasing

step-by-step explanation:

because its top point is at the to of the graph on the left side (atleast from the veiw i see it) now if that top point wast in the lower part of the graph it would be increasing hope this have a nice day