Construct a truth table for each of the following compound propositions.

a$)p^{{?}^{?}}$notp

b$)(pvvnotq)q$

c$)(pq)harr(notq$notp$)$

Write each of the following statements in the form $"$if $p,$ then $q^{\text{'}\text{'}}$ in English. $($Hint: Refer to the list of common ways to express implications listed in this section.$)$

a$.$ The newspaper will not come if there is an inch of snow on the street.

b$.$ It snows whenever the wind blows from the northeast.

c$.$ That prices go up implies that supply will be plentiful.

d$.$ It is necessary to read the textbook to understand the materials of this course.

e$.$ For a number to be divisible by $3,$ it is sufficient that it is the sum of three consecutive integers.

f$.$ Your guarantee is good only if you bought your TV less than $90$ days ago.

Write each of the following propositions in the form $"p$ if and only if $q^{\text{'}\text{'}}$ in English.

a$.$ If it is hot outside you drink a lot of water, and if you drink a lot of water it is hot outside.

b$.$ For a program to be readable it is necessary and sufficient that it is well structured.

C$.$ I like fruits only if they are fresh, and fruits are fresh only if I like them.

d$.$ If you eat too much sweets your teeth will decay, and conversely.

e$.$ The store is closed on exactly those days when I want to shop there.