Eight enable C2 be the code defined partially 2. (If you haven't but came upon the minimal distance in C2, achieve this now. ) utilizing the result of components five and seven, clarify why mistakes in any codeword can consistently be detected, and why one errors in any codeword can continuously be corrected. bankruptcy 4 effortless homes OF teams Is it attainable for a bunch to have diversified id parts? good, consider e1 and e2 are identification parts of a few staff G. Then e1 * e2 = e2 simply because e1 is an identification aspect, and e1 * e2 = e1 simply because e2 is an identification point for this reason e1 = e2 This exhibits that during each workforce there's precisely one identification point. Can a component a in a gaggle have diverse inverses! good, if a1 and a2 are either inverses of a, then a1 * (a * a2) = a1* e = a1 and (a1 * a)*a2 = e*a2 = a2 by means of the associative legislation, a1 * (a * a2) = (a1 * a) * a) * a2; therefore a1 = a2. This exhibits that during each team, every one point has precisely one inverse. during the past we've used the logo * to designate the crowd operation. different, traditionally used symbols are + and · (“plus” and “multiply”). whilst + is used to indicate the gang operation, we are saying we're utilizing additive notation, and we confer with a + b because the sum of a and b. (Remember and b wouldn't have to be numbers and for this reason “sum” doesn't, normally, seek advice from including numbers. ) while · is used to indicate the gang operation, we are saying we're utilizing multiplicative notation’, we frequently write ab rather than a-b, and make contact with ab the fabricated from a and b. (Once back, do not forget that “product” doesn't, as a rule, consult with multiplying numbers. ) Multiplicative notation is the preferred since it is straightforward and saves house. within the rest of this publication multiplicative notation should be used other than the place in a different way indicated. specifically, after we characterize a gaggle by means of a letter reminiscent of G or H, it will likely be understood that the group’s operation is written as multiplication. there's universal contract that during additive notation the id point is denoted by means of zero, and the inverse of a is written as −a. (It is termed the damaging of a. ) In multiplicative notation the id point is e and the inverse of a is written as a−1 (“a inverse”). it's also a practice that + is for use just for commutative operations. the main simple rule of calculation in teams is the cancellation legislations, which permits us to cancel the issue a within the equations ab = ac and ab = ca. it will be our first theorem approximately teams. Theorem 1 If G is a bunch and a, b, c are components of G, then (i) ab = acimpliesb = c and (ii) ba = caimpliesb = c you will see why this is often actual: if we multiply (on the left) either side of the equation ab = ac via a−1, we get b = c. when it comes to ba = ca, we multiply at the correct through a−1. this can be the belief of the facts; now here's the facts: half (ii) is proved analogously. quite often, we can't cancel a within the equation ab = ca. (Why now not? ) Theorem 2 If G is a bunch and a, b are components of G, then ab=eimpliesa=b−1andb = a−l The facts is particularly uncomplicated: if ab = e, then ab = aa−1 so via the cancellation legislation, b = a−1.