Assign a grade of A (correct), C (partially correct), or F (failure) to each.Justify assignments of grades other than A. (a) Claim. If and thenProof. If then by definition ofTherefore Thus(b) Claim. If and thenProof. Suppose Then Therefore we concludethat which proves the set inclusion.(c) Claim. If and is a family of subsets of A, thenProof. Suppose Then for y y f (Da) all a. Thus af (Da).af (Da) f QaDaR.f: A B {Da: a }z f 1( f (X)),z X. f (z) f (X).X f1 f: A B X A, ( f (X)).f 1 x X. ( f (X)) X.f f (x) f (X). 1 x f , 1( f(X)),f 1 f: A B X A, ( f (X)) X.4.6 Sequences 225there exists such that for all Then and xax Da f (x) = y, a. Daso Therefore,4.6

HIGHER-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS II: Nonhomogeneous Equations David Levermore Department of Mathematics University of Maryland 14 March 2012 Because the presentation of this material in lecture will diﬀer from that in the book, I felt that notes that closely follow the lecture presentation might be appreciated. 4. Nonhomogeneous Equations: General Methods and Theory 4.1. Particular and General Solutions 2 4.2. Solutions of Initial-Value Problems 3 5. Nonhomogeneous Equations with Constant Coeﬃcients 5.1. Unde