Question

In each of the following problems, three sets are described. One of the sets is not the same as the other two. In each case, find the set that is not like the others. (you do not need to prove, a very brief explanation is sufficient)

(a) A = {a + b | a ∈ N, b ∈ N}; B = {a − b | a ∈ N, b ∈ N}; C = N.

(b) A = {x^2 | x ∈ N}; B = {x^2 | x ∈ Z}; C = {x^2 | x ∈ R}.

(c) A = {4z + 2 | z ∈ Z}; B = {4z − 2 | z ∈ Z}; C = {2z + 4 | z ∈ Z}.

(d) A = {m | m ∈ Z ∧ m ≡ 4 (mod 3)};

B = {m | m ∈ Z ∧ m ≡ −4 (mod 3)};

C = {m | m ∈ Z ∧ m ≡ 14 (mod 3)}.

Answer 1

a) Here the given sets ame, A = {atbl aftl, ben? B = { ara lang bent عنيا see thay in the set A, ale verb ors be positive verber which are also nateral afh, ben 2 Also) in the ser e alt, ut itself being of National number, conlesing the set are natural rembers, But here the set alb, B is different os, it Conking poschre of wellas negalime numbers, as os for some a where aber, get regahne vermbers, so hone, sels are similar and B de arfforest. we and

by AB Ersetih хси", e=&r InER {uf Herrey in the set A As, all neth, So, the mergers be A of full nember Similarly of all and Z in B for be soll eleber. members in B will Buh in ci as Contains full VER, so they set as well a fraction nerober romber, .. The Set è is afferent from A, B. c) Az 842+2/ ZEZ? $ 43-%} .CZ c2 & 22 + 4 22 +4 / 2 62} Then for et 33, 2, 6, -2, 10, -6,1, -1°, 18, -14, ...3 3-2, 2, -6, 6; -10,10-14, ....3 $ 4, 6,218,0,19,-29-2-3 A = e 2 AB are Sindar er a de sarfferers as

AIB never Contain but C Contaning o. mez dy AS m = 4 (mod 3)} {mme meanm=-4 (med 3)} - Emmez 333 B m = 14 (mod this we see the elements in A in the Gurg m = 3 Kat 4.jKE T. the levert are in the formy 3 kg - 4 kot I. and clemens in @ are in the formy ma 343 ty, 2 KeEZ. thers the Similar, 'Be will be arsheret. bees A will be by cheeking elements, IE A boly 18 B, C also, 2 & A boy 2 EB, C.