MAT3702 ASSIGNMENT 2 2024
S2
DUE 14 AUGUST
, ASSIGNMENT 2
QUESTION 1
a) Closure under addition- For all x, y ∈ L, x + y ∈ L.
0+0=0∈L
0+2=2∈L
0+4=4∈L
2+2=4∈L
2+4=0∈L
4+4=2∈L
b) Closure under multiplication-For all x, y ∈ L, x * y ∈ L.
0*0=0∈L
0*2=0∈L
0*4=0∈L
2*2=4∈L
2*4=2∈L
4*4=4∈L
Therefore, L = {0, 2, 4} is a subring of ZZ6.
The identity element in a ring must satisfy the property that for all x ∈ L, x + 1 = x and x * 1 = x.
In L, the only element that satisfies this property is 0, as 0 + x = x and 0 * x = 0 for all x ∈ L.
Therefore, L has the identity element 0.
QUESTION 2
S2
DUE 14 AUGUST
, ASSIGNMENT 2
QUESTION 1
a) Closure under addition- For all x, y ∈ L, x + y ∈ L.
0+0=0∈L
0+2=2∈L
0+4=4∈L
2+2=4∈L
2+4=0∈L
4+4=2∈L
b) Closure under multiplication-For all x, y ∈ L, x * y ∈ L.
0*0=0∈L
0*2=0∈L
0*4=0∈L
2*2=4∈L
2*4=2∈L
4*4=4∈L
Therefore, L = {0, 2, 4} is a subring of ZZ6.
The identity element in a ring must satisfy the property that for all x ∈ L, x + 1 = x and x * 1 = x.
In L, the only element that satisfies this property is 0, as 0 + x = x and 0 * x = 0 for all x ∈ L.
Therefore, L has the identity element 0.
QUESTION 2
