After his lunch, Gdel tells you that he ate either apples or bananas; also that if he had corn, then he had a decaf coffee; also that he didn't drink either decaf coffee or espresso. Define propositions A through E for relevant parts of Gdel's statements. (For example, define A to be "Gdel ate apples" and so on.) Write each of the statements Gdel made about his lunch in terms of the propositions A through E. (Note that, since Gdel is a logician, he uses the word "or" to mean a logical OR, rather than the "or" used by everyday mortals.) It is not widely known, but Gdel lied about everything: every statement he said was false. Determine what Gdel had for lunch. Justify your answer.