Homework SourseConsider the sets A n elementof Z exist k elementof Z n 3
Consider the sets A = {n elementof Z | (exist k elementof Z) [n = 3k]}, B = {n elementof Z | (exist_i, j elementof Z) [n = 15i + 12j]}. Prove that A = B.![Consider the sets A = {n elementof Z | (exist k elementof Z) [n = 3k]}, B = {n elementof Z | (exist_i, j elementof Z) [n = 15i + 12j]}. Prove that A = B.Soluti](/WebImages/34/consider-the-sets-a-n-elementof-z-exist-k-elementof-z-n-3-1099976-1761580970-0.webp "Consider the sets A = {n elementof Z | (exist k elementof Z) [n = 3k]}, B = {n elementof Z | (exist_i, j elementof Z) [n = 15i + 12j]}. Prove that A = B.Soluti")
Solution
We have A = { 3k: k Z} and B = { 15i+12j: i, j Z}. Now, 15i + 12j =3(5i+4j) is a multiple of 3. Therefore B A . Further, since 5 and 4 are co-prime , and since their difference is 1, for every integer k Z , can be expressed as 5i+4j, where i, j Z , hence 3k = 3(5i +4j ) = 15i+12j. Thus, A B. Hence, A = B.
