Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Verify an equivalence relation and perform well-defined arithmetic on classes.
- Justify the conclusion "Z/~ has classes [0],[1],[2],[3]; [3]+[2]=[1]" using the stated assumptions.
Before you start
Divisibility and modular arithmetic.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Why does the relation a~b when 4 divides a−b partition the integers?
Why this math matters
Verify an equivalence relation and perform well-defined arithmetic on classes. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The underlying set is all integers.
- Class addition uses integer addition before taking a remainder.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Compress integers into residue classes
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Why does the relation a~b when 4 divides a−b partition the integers?
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
4|(a−a); 4|(a−b) ⇒ 4|(b−a)
Zero is divisible by four, and changing sign preserves divisibility: reflexivity and symmetry hold.
Work through the mathematics
4|(a−b), 4|(b−c) ⇒ 4|(a−c)
Adding the differences establishes transitivity.
Check the conclusion
Z/~ has classes [0],[1],[2],[3]; [3]+[2]=[1]
Every integer has one remainder, and changing representatives changes the sum by a multiple of four.
The result
Z/~ has classes [0],[1],[2],[3]; [3]+[2]=[1]
Every integer has one remainder, and changing representatives changes the sum by a multiple of four.
Common mistakes to catch
- An equivalence class is a set, not just its chosen label.
- Reflexivity and symmetry alone do not imply transitivity.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Which class contains −7?
Show a hint
Add a multiple of four.
Reveal answer and explanation
[1]
−7−1=−8 is divisible by four.
Practice 2
Why would a~b defined by |a−b|≤1 fail?
Show a hint
Test three consecutive integers.
Reveal answer and explanation
It is not transitive
0~1 and 1~2, but 0 is not related to 2.
Take the idea with you
Describe time-of-day addition using classes modulo 24.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Recover a greatest common divisor as a linear combination
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