Assign a grade of A (correct), C (partially correct), or F (failure) to each.Justify assignments of grades other than A.(a) Claim. If f is a one-to-one correspondence from A to B and g is a oneto-onecorrespondence from B to A, thenProof. Suppose f is a one-to-one correspondence from A to B and gis a one-to-one correspondence from B to A. Then by Theorem 4.4.1,is a one-to-one correspondence from A to A. Likewise, is aone-to-one correspondence from A to A. Then(b) Claim. If f is a permutation of A, thenProof. Since is the identity, Also is the identity, soTherefore(c) Claim. Let and be permutations of{1, 2, 3, 4, 5}. Then the inverse of is [3 2 4 1 5].Proof. Let and Then andTherefore(d) Claim. If f and g are permutations of A, thenProof. We know by Theorem 4.4.5 that is a permutation of A, sois a permutation of A. Also by Theorem 4.4.5, and arepermutations of A, so is a permutation of A. By Theorem 4.4.4,we can check whether is the inverse of by computing theircomposite. We find thatTherefore( f g) 1 = g1 f 1.

S343 Section 3.4 Notes- Repeated Roots of the Characteristic Equation; Reduction of Order 10-4-16 For + + = 0 and characteristic equation + + = 0, repeated roots occur when − 4 = 0 − o Follows...