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)

Identify the graph of y = ln x + 1.

Circle a has a diameter of 7 inches, a circumference of 21.98 inches, and an area of 38.465 square inches. the diameter of circle b is 6 inches, the circumference is 18.84 inches, and the area is 28.26 square inches. part a: using the formula for circumference, solve for the value of pi for each circle. (4 points) part b: use the formula for area and solve for the value of pi for each circle. (4 points)

Luke started a weight-loss program. the first week, he lost x pounds. the second week, he lost pounds less than times the pounds he lost the first week. the third week, he lost 1 pound more than of the pounds he lost the first week. liam started a weight-loss program when luke did. the first week, he lost 1 pound less than times the pounds luke lost the first week. the second week, he lost 4 pounds less than times the pounds luke lost the first week. the third week, he lost pound more than times the pounds luke lost the first week.