Suppose there are two goods and a consumer prefers more to less of each good. If the consumer has transitive preferences, can their difference curves intersect? Outline a short proof of your answer and illustrate with a diagram.

b. Blanche Carter has devised a system to grade men in her economics class. She cares about his intelligence and her appearance. She has rated each man on a scale of 1 to 5 for intelligence and 1 to 3 for looks. She defines a preference relation, R, as follows: xRy if child x scores at least as high as child y on appearance or intelligence. Give an example to show that Blanche's method of determining preferences might not lead to transitive preferences.