Question

Pe multimea numerelor reale se defineste legea de compositie
x+y=(2x-y+1)(2y-x+1)

a)Demonstrati ca legea de compozitie, este comutative.

b)Determinati perechile (m, n) de numere naturale pentru care
(2m) *n=13

Answer 1

x*y=(2x-y+1)(2y-x+1)

a. Comutativitatea

x*y=y*x

(2x-y+1)(2y-x+1)=(2y-x+1)(2x-y+1)

4xy-2x²+2x-2y²+xy-y+2y-x+1=4xy-2y²+2y-2x²+xy-x+2x-y+1

5xy-2x²-2y²+x+y+1=5xy-2x²-2y²+x+y+1 Adevarat⇒legea este comutativa

b. 2m*n=13

(4m-n+1)(2n-m+1)=13

Caz 1  1×13=13

4m-n+1=1

2n-2m+1=13

4m-n=0⇒ 4m=n

8m-2m=12

6m=12⇒ m=2

n=8

Caz 2

13×1=13

4m-n+1=13

2n-2m+1=1⇒ 2n=2m⇒ n=m

4m-m=12

3m=12

m=4

n=4