Consider the relation ~ on R given by x ~ y if and only if x = 2^n y for. some n
ID: 3108474 • Letter: C
Question
Consider the relation ~ on R given by x ~ y if and only if x = 2^n y for. some n Element Z. Prove that ~ is an equivalence relation. Suppose ~ is an equivalence relation on the set A and suppose a, b Element A. Prove that if [a] notequalto [b], then [a] Intersection [b] = 0. Let ~ be the equivalence relation on R given by x ~ y if and only if |x| = |y|. and let E be the set of equivalence classes. Suppose we define addition on E by the formula [x] + [y] = [x + y]. Prove or disprove that addition on E is well-defined. Suppose we define multiplication on E by the formula [x] middot [y] = [x middot y]. Prove or disprove that multiplication on E is well-defined. Write the addition and multiplication tables for Z_6. Suppose [a], [b] Element Z_6 and [a] middot [b] = [0]. Is it necessarily true that [a] = [0] or [b] = [0]? Use modular arithmetic to prove that an integer (written in base 10) is divisible by 9 if and only if the sum of its digits is divisible by 9. Find the multiplicative inverse of [13] in Z_60.Explanation / Answer
60=13*4+8 , 8=60-13*4
13=8+5 , 5=13-8=13*5-60
8=5+3 , 3=8-5=60*2-13*9
5=3+2 , 2=5-3=13*14-60*3
3=2+1 , 1=3-2=60*5-13*23
1=60*5-13*23
Hence, inverse of 13 is -23=37