Final answer:
In the algebraic structure Z₆, the statement '4*4=4' is not true because when you multiply 4 by 4 and take the result modulo 6, you get 16 mod 6, which is 4, not equal to 16 modulo 6.
Step-by-step explanation:
The question pertains to group theory within abstract algebra, specifically the properties of the algebraic structure Z₆, which is the set of integers modulo 6. This structure is equipped with addition and multiplication operations, where results are taken modulo 6.
Considering the statements provided:
- A. 4+4=2 is true, since adding 4 and 4 gives 8, and modulo 6, we get 2.
- B. a*b =0 but both a or b are not equal to zero is not necessarily true. This would imply the existence of zero divisors in Z₆, which is possible since 6 is not a prime number.
- C. -2=4 is true since adding -2 to 6 gives 4, which is the additive inverse of 2 in Z₆.
- D. a+0=a for every a is true, which is the identity property of addition.
- E. for every a there exists a b, such that a+b=0 is true, presenting the concept of additive inverses in Z₆.
- F. 1*a=a for every a is true, showing the multiplicative identity property.
- G. 4*4=4 is not true; 4 multiplied by 4 is 16, and mod 6 gives us 4, not equal to 16 modulo 6.
- H. 1+1+1+1+1+1=0 is true, as adding 1 six times is 6, and modulo 6, we return to 0.
- I. 11*7 = -1 = 5 is not a clear statement but typically in Z₆, 11*7 mod 6 would indeed result in 5.
- J. a*b =0 means a or b has to be equal to zero is not true in Z₄ as zero divisors are allowed. Here, the statement does not hold as neither a nor b is required to be zero for their product to be zero mod 6.
Therefore, the statement that is not true in Z₆ is G. 4*4=4.