8, Prove that if R is an integral domain and a E R such that a2 + 2a + 1-0 then a =-1. Then give an example of a ring that is not an integral domain for which a2a 1-0 but a f -1.
8, Prove that if R is an integral domain and a E R such that a2 + 2a + 1-0 then a =-1. Then give an example of a ring that is not an integral domain...
e) Use the triangle inequality to prove that (ac + bd)2 (a2 + b2)(c2 + d2) for all a, b, c, d e R. Total: [20 marks]
e) Use the triangle inequality to prove that (ac + bd)2 (a2 + b2)(c2 + d2) for all a, b, c, d e R. Total: [20 marks]
Prove for any two sets, E and F, E∪F =E∪(Ec ∩F) Be sure to justify every statement you make by referring back to your definitions.
5. Prove that if A1, A2, ... An and B are sets, then (A. – B) U (A2 – B) U... U (An – B) = (A, U A, U... U An) – B.
(D) Does the pollt1lic Recall that for every vector a R? we defined the translation, Prove that for any vectors a, a2 E R2 nidne have the measures
(D) Does the pollt1lic Recall that for every vector a R? we defined the translation, Prove that for any vectors a, a2 E R2 nidne have the measures
3. [5 pts] Suppose A and A2 are disjoint. Prove that
Given f, g E Da,b] , prove that min(f, g) E Da,시
Let A be a square matrix. Prove that if A2 = A, then I - 2A is the inverse of I - 2A.
(4) w e suppose j is a measure, f E L1(μ),dom(f)-R, f 0 and EnaER: f(x) 2 n). a) Prove that limn-100 JE, fdy -0 (b) Prove that limn-0o n(En)-0
(4) w e suppose j is a measure, f E L1(μ),dom(f)-R, f 0 and EnaER: f(x) 2 n). a) Prove that limn-100 JE, fdy -0 (b) Prove that limn-0o n(En)-0
6. Prove that if a and b are odd integers, then a2 is divisible by 8. 7. Prove that if a is an odd integer, then ta + (a + 2)?+ (a +4)2 +1) is divisible by 12.