Question

Question:In Exercise 13-15, K is a subgroup of G . Determine whether the given cosets are disjoint or identical.

13. $\mathit{G}\mathbf{=}\mathbf{\mathbb{Z}}$;$\mathit{K}\mathbf{=}<7>$ (a) K+4 and K+3 .

Short answer

The given cosets are disjoint

Step-by-step solution

Step by Step Solution Step 1: Subgroup K of G  

Given that K is a cyclic subgroup of the additive group Z.

The subgroup K consists of all the multiples of 7 and the cosets of K are the congruence classes modulo 7.

The cosets are disjoint.

The coset K+4 consists of all integers leaving a remainder 4 when divided by 7 and the coset K+3 consists of all integers leaving a remainder 3 when divided by 3.

Since, $3\notin K+4$implies that the elements in K+3 does not belong to set K+4 .

Thus, the intersection of two cosets are empty.

Therefore, the given cosets K+4 and K+3 are disjoint.