Question

(a) Prove that any line through the origin of the xy-plane (a.k.a., $\mathbb{R}^2$) may be regarded as a subgroup of $\mathbb{R}^2$. [Hint: A line of slope $m$ may be described as $L_m = \{(x, y) \in \mathbb{R}^2 | y = mx\}$.] (b) Prove that $L_m \cong \mathbb{R}$ for any $m$.

          (a) Prove that any line through the origin of the xy-plane (a.k.a., $\mathbb{R}^2$) may be regarded as a subgroup of $\mathbb{R}^2$. [Hint: A line of slope $m$ may be described as $L_m = \{(x, y) \in \mathbb{R}^2 | y = mx\}$.]
(b) Prove that $L_m \cong \mathbb{R}$ for any $m$.

$(a) Prove that any line through the origin of the xy-plane (a.k.a., ℝ^2) may be regarded as a subgroup of ℝ^2. [Hint: A line of slope m may be described as Lm = {(x, y) ∈ℝ^2 | y = mx}.] (b) Prove that Lm ≅ℝ for any m.$

Added by Matthew G.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Ankit Gupta

07/29/2020

Step 1

Step 1: A subgroup of R2 is a subset of R2 that is closed under addition and contains the additive inverse of each of its elements. Show more…

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a) Prove that any line through the origin of the xy-plane, also known as R2, may be regarded as a subgroup of R2. [Hint: A line of slope m may be described as Lm={xy}∈IR|y=mx}.] b) Prove that LmR for any m.