Welcome to CS103A!



—> you will learn here why thing are true , not check theat 






Suppose you improve at some skill at rate of “1%” per day , How much better at that skill will you be by the end of the yeat 
After one day , youve 1.01 times better 
After two days , you are (1.10)2 times better 

After one year , you’lle 
(1.01)365 = 37.8 times better 

→ this what we called compound affect righ ? 

Yes, this is exactly what's called the compound effect! You've understood it perfectly. Let me break down this fascinating concept:

What you're describing is compound growth, similar to compound interest in finance, but applied to personal growth and skill development. The key insight is that small, consistent improvements compound over time to create dramatic results.

Let's verify your math:
- Day 1: 1.01¹ (1.01 times better)
- Day 2: 1.01² (1.0201 times better)
- Day 365: 1.01³⁶⁵ ≈ 37.8 times better

Your calculation is correct! This demonstrates why the saying "1% better every day" is so powerful. Even though 1% seems tiny, over a year it leads to massive improvement.

However, there's one important caveat: In real skill development, improvement usually isn't this mathematically perfect. Progress often comes in bursts and plateaus. But the underlying principle remains valid - small, consistent efforts compound over time to create remarkable results.

Would you like to explore how this concept applies to specific skills or goals?




Donc el souel honi enik 3andik zouz sets ‘A’ and la5er “B” , 
Kefh bsh tproof enou one set is a subset of another ? 


→ dema n
E7eb nbda el logic mt3ik eniik nts3ml exmple sghir oky concrete and b3ed abstract or distilled to make it more general approach oky 

Let’ say A= {2,-8,7} or B={2,5,6,7,-8,11} 

O el objective honi elik tbyen enou A howa subset men el Set B  oky : 

—> ntswer awl fekra jet le mo5i honi enik t7welll tbyen enou kol element fe el Set A mwjoud fe set B 

Y3ni fe tari9a o5ra tw7ell tbyen enou kol mt5o element men set A bsh tal9h mwjoud be set B oky 

Fe example mt3na honii A ken n54o 2 bsh  nl9wh majoud fe B 
Kef kef ken 54et -8 bsh tnl9ah mjoud fe B 
Kef kef ken 54det 7 bsh tl9h mjoud fe B okiii 

Donc n7ebk awll 7aje tw7li el tounsi mt3i en englais mt8er mtbdell the style mt3i oki just grammer oky 

So, let’s say you have two sets, one called ‘A’ and the other one ‘B’. How would you prove that one set is a subset of the other?
→ Let’s assume the size is n.
I like to start with concrete, small examples and then generalize the logic. So, let’s use a specific example first, then abstract the approach to make it more general.
Let’s say A = {2, -8, 7} and B = {2, 5, 6, 7, -8, 11}.
The objective here is to show that A is a subset of set B, okay?
→ So, the first thought that came to my mind is to prove that every element in set A exists in set B.
This means, in other words, you would show that for each element you pick from set A, you will find it present in set B, okay?
In our example, if you pick the number 2 from A, you’ll find it present in B.
Similarly, if you pick -8, you’ll also find it present in B.
And if you pick 7, you’ll find it in B too, okay?

Does that flow sound good? Let me know if you want any changes!
4o


Q- so now you have my tunisia logic and english version give a good feabacka here in english 

Q- be my math mentor do not give a solutio direcly just try to builde my logic going first form my thinking process oky 

Here’s the formatted feedback and revised approach:

---

**Your approach to proving that one set is a subset of another is clear and well-structured. Here's some feedback on your English version to enhance clarity and precision:**

### **Original English Version:**

So, let’s say you have two sets, one called ‘A’ and the other one ‘B’. How would you prove that one set is a subset of the other?

→ Let’s assume the size is n.

I like to start with concrete, small examples and then generalize the logic. So, let’s use a specific example first, then abstract the approach to make it more general.

Let’s say A = {2, -8, 7} and B = {2, 5, 6, 7, -8, 11}.

The objective here is to show that A is a subset of set B, okay?

→ So, the first thought that came to my mind is to prove that every element in set A exists in set B.

This means, in other words, you would show that for each element you pick from set A, you will find it present in set B, okay?

In our example, if you pick the number 2 from A, you’ll find it present in B.

Similarly, if you pick -8, you’ll also find it present in B.

And if you pick 7, you’ll find it in B too, okay?

---

### **Revised Version:**

Let’s say you have two sets, A and B. How would you prove that one set is a subset of the other?

I prefer to start with small, concrete examples and then generalize the logic. So, let's use a specific example first and then abstract the approach to make it more universally applicable.

For instance, let A = {2, -8, 7} and B = {2, 5, 6, 7, -8, 11}.

The goal here is to demonstrate that set A is a subset of set B.

The initial idea is to show that every element in set A exists in set B.

In other words, for each element you select from set A, you should be able to find it in set B.

In our example:
- If you pick the number 2 from A, you'll find it in B.
- Similarly, -8 from A is present in B.
- And 7 from A is also found in B.

Therefore, set A is indeed a subset of set B.

---

### **Feedback Highlights:**
1. **Clarity and Conciseness:** The revised version aims for clearer sentence structures and removes conversational fillers like “okay?”, replacing them with more formal language.
2. **Logical Flow:** Each step logically follows the previous one, making the argument easy to follow.
3. **Consistency:** Ensuring consistent terminology, such as consistently referring to "set A" and "set B" without unnecessary repetition.
4. **Precision:** Clearly stating the objective and the method to achieve it enhances the mathematical rigor of the explanation.

---

### **Building Your Logical Approach**

Let’s further develop your logic to prove that set A is a subset of set B. Here’s a guided approach based on your initial thinking:

1. **Define the Sets Clearly:**
   - Set A: {2, -8, 7}
   - Set B: {2, 5, 6, 7, -8, 11}

2. **State the Objective:**
   - **Objective:** Prove that A is a subset of B (denoted as A ⊆ B).

3. **Understand the Definition:**
   - **Subset Definition:** Set A is a subset of set B if every element of A is also an element of B.

4. **Develop the Proof Strategy:**
   - **Method:** Verify that each element in A is present in B.

5. **Execute the Proof:**
   - Element 1: 2 ∈ A. Check if 2 ∈ B.
   - Element 2: -8 ∈ A. Check if -8 ∈ B.
   - Element 3: 7 ∈ A. Check if 7 ∈ B.

6. **Conclude:**
   - Since all elements of A are found in B, A is indeed a subset of B.

---

### **Generalizing the Approach**

After working through specific examples, you can generalize the proof for any two sets A and B:

1. **Assume:** A and B are any two arbitrary sets.
2. **Condition:** To prove A ⊆ B, take an arbitrary element x where x ∈ A.
3. **Show:** Demonstrate that x ∈ B.
4. **Conclusion:** Since x was arbitrary, this holds for all elements in A, thus A ⊆ B.

---

### **Tips for Effective Proofs:**
1. **Start with Definitions:** Always begin by stating relevant definitions to ensure clarity.
2. **Use Arbitrary Elements:** When generalizing, use arbitrary elements to make your proof universally applicable.
3. **Logical Progression:** Ensure each step follows logically from the previous one.
4. **Conclude Clearly:** Summarize the findings to reinforce that the objective has been met.

Feel free to ask more questions or request further clarification on any of these steps!

Q-now use mit lelve mathemtcial formale notationa and lanage to proof this fro every set oky