(1.1.8 in the book) Let P be the proposition ’I bought a lottery ticket this weekend’ and Q be the proposition ’I won the million dollar jackpot’. Express each of the following propositions in english. a) ¬P b) P ∨ Q c) P → Q d) P ∧ Q e) P ⇐⇒ Q f) ¬P → ¬Q g) ¬P ∧ ¬Q h) ¬P ∨ (P ∧ Q) - can this one be simplified at all?

~ is for negation

^ is for "and"

v is for "or"

=> for "if then"

<=> for "if and only if"

Step-by-step explanation:

(a) ~P (negation of P)

I didn't buy a lottery ticket this weekend.

(b) P v Q (P is in disjunction Q)

I have either bought a lottery ticket this weekend or won the million dollar jackpot.

(c) P => Q (Q is a consequence of P)

I won the million dollar jackpot because I bought a lottery ticket this weekend.

(d) P ^ Q (P is in conjunction with Q)

I bought a lottery ticket this weekend, and I won the million dollar jackpot.

(e) P <=> Q (P and Q are dependent on each other)

If only I had bought a lottery ticket this weekend, I would have won the million dollar jackpot.

(f) ~P => ~Q (negation of Q is a consequence of negation of P)

I didn't win the million dollar jackpot because I didn't buy a lottery ticket this weekend.

(g) ~P ^ ~Q (negation of P is in conjunction with negation of Q)

I neither bought a lottery ticket this weekend nor won the million dollar jackpot.

(h) ~P v (P ^ Q)

This is logically equivalent to (~P v P) ^ (~P v Q) (negation of P is in disjunction with P, and also with disjunction with Q), and can be best expressed as:

It didn't matter that I bought a jackpot ticket or not, I won the million dollar jackpot.

a) 0,0; 1,1; and 2,2

step-by-step explanation:

these have the exact same coordinates, thus they are the correct function.

answer: the sun

explanation:

water cycle describes the evaporation of water from earth's surface, formation of clouds and precipitation water back onto the surface of earth. seventy percent of the earth is covered with water in the form of seas and oceans. the heat of the sun causes the evaporation of water from the surface of the ocean. the water vapors rise and condense to form water droplets. the collection of these water droplets form clouds which rain down again.

the evaporation of the oceans surface water as part of the water cycle is caused by the sun