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 Z^{n+1} which is a countable set.

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