To prove: D+⊆Q+ Given information: Each x ∈D is identified with [x,e]in Q

Question: "To prove: D+⊆Q+ Given information: Each x ∈D..."

High School
Geometry
High school geometry
Transformation properties

Sinead Mcgee

Answered question

2020-11-27

To prove:

D^{+} \subseteq Q^{+}

Given information:

Each x \in D is identified with [x, e] in Q

Answer & Explanation

sovienesY

Skilled2020-11-28Added 89 answers

Formula used:
1) An ordered field is an ordered integral domain that is also a field.
2) In the quotient field Q of an ordered integral domain D, defined $Q^{+}$ by
$Q^{+} = {[a, b] ∣ a b \in D^{+}}$
3) Well-ordered $D^{+} :$
If D is an ordered integral domain in which the set $D^{+}$ of positive elements is well-ordered, then e is the least element of $D^{+} .$
Proof:
$Let x \in D^{+}, then x \in D$
$So, [x, e] in Q where e is the least element of D^{+}$
$Now e \in D^{+} and x \in D^{+} implies that x e \in D^{+}$
$Therefore, [x, e] \in Q^{+} \subseteq Q$
$So, x \in Q^{+}$
$Hence, D^{+} \subseteq Q^{+}$

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