Please answer b & c. Please justify all work and explain in good detail how to o
ID: 3168223 • Letter: P
Question
Please answer b & c.
Please justify all work and explain in good detail how to obtain each answer. The table with #s1-14 are what properties we are determining that hold or not. 1. For each of the number systems below determine whether each proper justification for each answer. Then fill in the table with N's and Y's to (a) 3Z = { . . . ,-6,-3, 0, 3, 6, . . .} is the set of all multiples of 3 with t ion (b) Q+U[0) is the set of all non-negative rational numbers with the u (c) R = {[0] 10, [2] 10, [4] 10, [6]10, [8]io) c z10. (d) S = Z with the operations of(addition) and O (multiplication) a@b=ab _ 1. Property 1. The system is closed under addition. 2. The system is closed under multiplication. 3. Addition is associative. 4. Multiplication is commutative. 5. Multiplication distributes over addition. 6. There is an additive identity. . Each element has an additive inverse. 8. There is a multiplicative identity. 9. Every nonzero element has a multiplicative inverse. 10. The zero product property of multiplication holds 11. There are no zero divisors. 12. Additive Cancellation holds 13. Multiplicative cancellation (of nonzero elements) holds 14. The ordering axioms hold.Explanation / Answer
b) 1.the set of non negative rational numbers are closed under addition since sum of two nonnegative rational number is again a nonnegative rational number.
2. product of two nonnegative rational number is again a nonnegative rational number so true
3. Sum of any three nonnegative rational number is associative as a+(b+c)=(a+b)+c for all a,b,c in Q+U{0}
4. Multiplication of nonnegative rational number is commutative as ab=ba
5. multiplication is distributive over addition it is true as a(b+c)=ab+ac
6. 0 is the additive identity of the set
7. each element does not have an additive inverse for example 2 is in the set but not -2.
8. 1 is the multiplicative identity
9. each element has a multiplicative inverse since if a is in the set then 1/a is also in the set.
10. yes on multiplying with zero the result is zero
11. yes because if ab=0 then a or b=0
12. yes additive cancellation holds since if a+b=a+c then b=c
13. yes as if ab=ac then a=c
14. yes since for each rational number there is always a rational number greater than that.
c)1. yes since sum of two even number will be again an even number so that their remainder by modulo 10 will be in the R only
2. yes since multiplication of two even numbers is again an even number.
3. modulo addition is always associative
4. modulo multiplication is always associative
5. modulo multiplication is distributive over addition
6. 0 is the additive identity
7. yes 0 is self inverse, 2 has inverse 8, 4 has inverse 6.
8. 6 is multiplicative identity since 0*6=0, 2*6=2, 4*6=4 and 8*6=8
9. yes since 2 has inverse 8, 6 is self inverse and 4 is also self inverse
10. yes it holds since 0a=0
11. yes because any two nonzero numbers dont give zero on product
12. yes since R is an additive group
13. yes since R is a multiplicative group as well
14. not sure
2.