II Set Theory 14 points 1 Let C n Zn6r5forsomerZ and D mZ

II. Set Theory (14 points)

(1)

Let C = { n Z|n=6r5forsomerZ} and D = {mZ | m=3s+1 for some s Z}.

Prove or disprove each of the following:

(a) C D (b) D C

(2)

Use an “element argument”1 to prove the following statements for any sets A and B:

(a) ABBc Ac

(b) (AB)(BA)(AB)=AB

(3)

The symmetric difference of two sets S and T is the set of objects that are in exactly one of the sets. It is denoted by ST, and formally defined as ST = (S T) (T S).

Prove or disprove the following identities about the symmetric difference of sets:

(a) A(BC) = (AB)C

(b) A(BC)=(AB)C

Hint: You may want to draw Venn diagrams to help build the intuition behind the sym- metric difference operator.

Solution

only one question allowed per submission.

a) to prove this we need to show that for every x belonging to C we should show that x doesnot belong to D

let x be an elemnt of C then , there exist r such that

x =6r-5

x = 6r -6+1

x =3*(2r-2)+1

x =3s+1 where s = 2r-2

since r is integer s alsois integer

thus we have proved element of C is in D

b) to prove this we need to show that for every x belonging to D we should show that x doesnot belong to C

let x be an element of D then x = 3s+1

it can be seen that x cannot be expressed as x =6r-5, thus there element where x belong to D but not toC

hence proved