三至五年级Category Theory计划中
One thing after another
Doing one action and then another: chains of actions on shapes, numbers and objects, how the order can matter, how to undo a chain by undoing its steps in reverse order, and arrow pictures showing how one thing leads to another.
math.cat.chaining-actions
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前置知识
1 项必需的前置知识（一直追溯到底）
你将学会
- Carry out chains of actions such as 'add 5, then double' and predict their results.
- Find examples where swapping the order of two actions changes the result, and examples where it does not.
- Undo a chain of actions by undoing its steps in reverse order, as with socks and shoes.
- Draw arrow pictures showing how objects are related or transformed, and follow paths of arrows.
定义
- Chaining two actions means doing the first and then doing the second to the result.
- An arrow picture shows objects as dots and actions or relationships as arrows from one dot to another.
定理
- 'Add 5, then double' and 'double, then add 5' usually give different results.
- To undo 'put on socks, then put on shoes', take off the shoes first and then the socks.
- Doing 'turn a quarter turn' four times in a row has the same effect as doing nothing.
常见误解
- Assuming the order of two actions never matters.
- Undoing a chain by undoing its steps in the original order.
通向
可选发现
- Composition Chaining actions becomes composition of functions g ∘ f, and undoing a chain becomes the rule (g ∘ f)⁻¹ = f⁻¹ ∘ g⁻¹.
- Groups The socks-and-shoes rule (gh)⁻¹ = h⁻¹g⁻¹ holds in every group.
- ← Following and giving precise instructions
- ← Expressions, unknowns and rules
- ← Symmetry: slides, turns and flips