Abstract Algebra-1-Binary Operations, guaranteed and verified 100% Pass

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Binary Operations
Binary operations are important, in part, because they are used in the definitions
of groups, rings, and fields.



Def: A binary operation ∗ on a set S is a function mapping 𝑆 × 𝑆 into 𝑆. For
each (𝑎, 𝑏) ∈ 𝑆 × 𝑆 , ∗ (𝑎, 𝑏) ∈ 𝑆.



Ex. Addition and multiplication are both binary operations on
ℤ, ℝ, ℂ, ℝ+ , 𝑜𝑟 ℤ+ (ℝ+ = {𝑥 ∈ ℝ| 𝑥 > 0}, similarly for ℤ+ ).
+∶ ℝ × ℝ → ℝ
(𝑎, 𝑏) → 𝑎 + 𝑏 i. e. + (𝑎, 𝑏) = 𝑎 + 𝑏 ∈ ℝ
∙ ∶ ℝ×ℝ→ℝ
(𝑎, 𝑏) → 𝑎 ∙ 𝑏 i. e. ∙ (𝑎, 𝑏) = 𝑎 ∙ 𝑏 ∈ ℝ.


Ex. Division is not a binary operation on ℤ, ℤ+ , 𝑜𝑟 ℝ.

1. It’s not a binary operation on ℤ because
÷∶ ℤ × ℤ → ℤ
𝑎
(𝑎, 𝑏) →
𝑏
𝑏 ≠ 0 thus ÷ is not defined for all points in ℤ × ℤ.

2. It’s not a binary operation on ℤ+ because
÷∶ ℤ+ × ℤ+ → ℤ+
𝑎
(𝑎, 𝑏) →
𝑏
is not defined for points where 𝑏 doesn’t divide 𝑎
2
(e.g. 𝑎 = 2, 𝑏 = 3, ∉ ℤ+ ).
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3. It’s not a binary operation on ℝ because
÷∶ ℝ × ℝ → ℝ
𝑎
(𝑎, 𝑏) → is not defined when 𝑏 = 0.
𝑏

Notice that ÷ is a binary operation on

ℝ∗ = ℝ − {0}, ℝ+ , ℚ∗ = ℚ − {0}, and ℚ+ .


Def: Let ∗ be a binary operation on 𝑆 and let 𝐻 be a subset of 𝑆. The
subset 𝐻 is closed under ∗ if for all 𝑎, 𝑏 ∈ 𝐻, 𝑎 ∗ 𝑏 ∈ 𝐻.



Ex. + is a binary operation on ℝ but + is not a binary operation on

ℝ∗ = ℝ − {0} because
+∶ ℝ∗ × ℝ∗ → ℝ∗
(𝑎, 𝑏) → 𝑎 + 𝑏
+(1, −1) = 0 ∉ ℝ∗ .


Ex. + and ∙ are binary operations on ℤ.



Ex. Let 𝐻 = {2𝑛 − 1 ⃒ 𝑛 ∈ ℤ+ } = {1, 3, 5, 7, 9, … } ⊆ ℤ+

Determine whether
a) H is closed under +
b) H is closed under ∗.