Questions 20 & 22 Pre-Write

22. Let (R,0,1,+,⋅,≤) be an ordered ring with identity.
Statement: If x>0, then −x<0.
Proof:
x + (-x) = 0 is the definition of an inverse
x>0
x + (-x) > 0 + (-x) add -x to both sides
0 > -x by the definition of an inverse and the additive property of 0

Statement: 1>0.
Proof:
1·n=n by the identity property
1>0
1·n>0·n multiply by n on both sides
n>0 by the identity property and the multiplicative identity of zero

20. Statement: The polynomial ring Z[x] over the integers is countably infinite.
Proof:
-show that the union of countable sets is countable (haven't learned yet)
-Z[x] is the union of all polynomials of degree i, if i∈N.
-the countable union of Zn+1 which is a countable set.

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