Question

Vă rog frumos să mă ajutați! Urgent!​

Answer 1

Explicație pas cu pas:

a)

x ○ y = 2xy - 2x - 2y + 3

y ○ x = 2yx - 2y - 2x + 3

=> x ○ y = y ○ x

=> legea "○" este comutativă

x ○ (y ○ z) = 2x(y ○ z) - 2x - 2(y ○ z) + 3 = 2x(2yz - 2y - 2z + 3) - 2x - 2(2yz - 2y - 2z + 3) + 3 = 4xyz - 4xy - 4xz + 6x - 2x - 4yz + 4y + 4z - 6 + 3 = 4xyz - 4xy - 4xz - 4yz + 4x + 4y + 4z - 3

(x ○ y) ○ z = 2(x ○ y)z - 2(x ○ y) - 2z + 3 = 2(2xy - 2x - 2y + 3)z - 2(2xy - 2x - 2y + 3) - 2z + 3 = 4xyz - 4xz - 4yz + 6z - 4xy + 4x + 4y - 6 - 2z + 3 = 4xyz - 4xz - 4yz - 4xy + 4x + 4y + 4z - 3

=> x ○ (y ○ z) = (x ○ y) ○ z

=> legea "○" este asociativă

b)

x ○ y = xy - 2x - 2y + 6

y ○ x = yx - 2y - 2x + 6

=> x ○ y = y ○ x

=> legea "○" este comutativă

x ○ (y ○ z) = x(y ○ z) - 2x - 2(y ○ z) + 6 = x(yz - 2y - 2z + 6) - 2x - 2(yz - 2y - 2z + 6) + 6 = xyz - 2xy - 2xz + 6x - 2x - 2yz + 4y + 4z - 12 + 6 = xyz - 2xy - 2xz - 2yz + 4x + 4y + 4z - 6

(x ○ y) ○ z = (x ○ y)z - 2(x ○ y) - 2z + 6 = (xy - 2x - 2y + 6)z - 2(xy - 2x - 2y + 6) - 2z + 6 = xyz - 2xz - 2yz + 6z - 2xy + 4x + 4y - 12 - 2z + 6 = xyz - 2xy - 2xz - 2yz + 4x + 4y + 4z - 6

=> x ○ (y ○ z) = (x ○ y) ○ z

=> legea "○" este asociativă

c)

x ○ y = x + y + xy

y ○ x = y + x + yx

=> x ○ y = y ○ x

=> legea "○" este comutativă

x ○ (y ○ z) = x + (y ○ z) + x(y ○ z) = x + (y + z + yz) + x(y + z + yz) = x + y + z + xy + xz + yz + xyz

(x ○ y) ○ z = (x ○ y) + z + (x ○ y)z = (x + y + xy) + z + (x + y + xy)z = x + y + xy + z + xz + yz + xyz = x + y + z + xy + xz + yz + xyz

=> x ○ (y ○ z) = (x ○ y) ○ z

=> legea "○" este asociativă