Assign a grade of A (correct), C (partially correct), or F (failure) to each. Justify assignments of grades other than A. (a) Claim. For every indexed family {A:},AA. 'Proof.' Choose any A{A:}. Then, by Theorem 2.3.2, AA and AA. Therefore, by Theorem 2.1.1(c), AA (b) Claim. If AB for all , then AB. 'Proof." Suppose xA. Then, because AB for all , xB. Therefore AB. c) Claim. For every indexed family {A:},AA. 'Proof.' Let ={r,s,t},Ar={a,b,c,d},As={b,c,d,e}, and At={c,d,e,f}. Then A={c,d}{a,b,c,d,e,f}=A