140/METHODS OF PROOF FOR BOOLEAN LOGIC Exercises In the following exercises, decide whether the displayed argument is valid. If it is, turn in an infor- mal proof, phrased in complete, well-formed English sentences, making use of first-order sentences as convenient. Whenever you use proof by cases or proof by contradiction, say so. You don't have to be explicit about the use of simple proof steps like conjunction elimination. If the argument is invalid, con- struct a counterexample world in Tarski's World. (Argument 5.16 is valid, and so will not require a counterexample.) 5.15 b is a tetrahedron. c is a cube. Either c is larger than b or else they are identical. b is smaller than c. 5.16 Max or Claire is at home but either Scruffy or Carl is unhappy. Either Max is not home or Carl is happy. Either Claire is not home or Scruffy is unhappy. Scruffy is unhappy.