Determine whether formulas u and v are logically equivalent (you may use truth tables or properties of logical equivalences).
(a) u : (p ⇒ q) ∧ (p ⇒ q) v : p
(b) u : p ⇒ q v : q ⇒ p
(c) u : p ⇔ q v : q ⇔ p
(d) u : (p ⇒ q) ⇒ r v : p ⇒ (q ⇒ r)

Write an equation for the function that includes the points (1,4/5) and (2,2/3)

1- is the product of two rational numbers irrational or rational? first, make a hypothesis by multiplying two rational numbers. then, use variables such as x=a/b and y=c/d and the closure property of integers to prove your hypothesis. 2- what do you think the product of a nonzero rational number and an irrational number is? is it rational or irrational? make use of variables, the closure property of integers, and possibly a proof by contradiction to prove your hypothesis. 3- why do we have to specify that the rational number must be nonzero when we determine what the product of a nonzero rational number and an irrational number is? if the rational number were 0, would it give us the same result we found in part b?

Avery’s piggy bank has 300 nickels, 450 pennies, and 150 dimes. she randomly picks three coins. each time she picks a coin, she makes a note of it and puts it back into the piggy bank before picking the next coin.