Скачать или смотреть Direct Proof: Square of an Odd Integer is Always Odd | VTU BCS405A Discrete Mathematics

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👉 BCS405A Module 1 – Fundamentals of Logic | VTU Dec 2024, June 2024, Model Papers Solved
🔗    • BCS405A Module 1 – Fundamentals of Logic |...  



🧠 Statement to Prove (Using Direct Proof):
"The square of an odd integer is an odd integer."

✍️ What is a Direct Proof?
A direct proof starts by assuming the hypothesis is true and uses definitions, axioms, and logical reasoning to prove that the conclusion is also true.

🔢 Step-by-Step Explanation (Direct Proof)
Step 1: Define an odd integer.
An integer is odd if it can be written in the form:
  n = 2k + 1,
where k is an integer.

Let’s assume n is an odd integer, so:
  n = 2k + 1 for some integer k.

Step 2: Square both sides.

Now compute the square of n:
  n² = (2k + 1)²
    = (2k)² + 2 × 2k × 1 + 1²
    = 4k² + 4k + 1

Factor 2 out of the first two terms:
  n² = 2(2k² + 2k) + 1

Now let m = 2k² + 2k, which is an integer.
Then:
  n² = 2m + 1

Step 3: Conclusion
Since n² = 2m + 1, and m is an integer,
it matches the definition of an odd number.

✅ Therefore, the square of an odd number is also an odd number.

📌 Why is this important for exams?
It’s a classic proof question under Direct Proofs in Module 1: Fundamentals of Logic.

It strengthens your understanding of mathematical logic, integers, and proof construction, all of which are essential in Discrete Mathematics and Computer Science.

🧠 Use in Programming & Math:
This concept helps in algorithm design, such as checking parity.

Used in number theory, cryptography, hashing, etc.

🔢 Video List:

1️⃣ Tautology Proof Explained | [(P ∧ ¬Q) → R] → [P → (Q ∨ R)] | VTU DMS BCS405A Module 1 Q1A
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2️⃣ Important Tautology Question for VTU Exams | Truth Table Proof | BCS405A Dec/Jan 2024-25 Q1(a)
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3️⃣ Prove Tautology by Truth Table | [(P ∨ Q) ∧ (P → R) ∧ (Q → R)] → R | BCS405A Module 1 Q1(a)
🔗    • Prove Tautology by Truth Table | [(P ∨ Q) ...  

4️⃣ Logical Equivalence Using Truth Table | VTU BCS405A | Q1(a) Model QP 1
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5️⃣ (P ∨ Q) → R ≡ (P → R) ∧ (Q → R) | Logical Equivalence Proof | VTU BCS405A Dec 2024 Q2(a)
🔗    • (P ∨ Q) → R ≡ (P → R) ∧ (Q → R) | Logical ...  

6️⃣ Laws of Logic Problem | VTU BCS405A Module 1 – Q.No. 2(b) | June-July 2024
🔗    • Laws of Logic Problem | VTU BCS405A Module...  

7️⃣ Prove [¬P∧(¬Q∧R)]∨[(Q∧R)∨(P∧R)] ≡ R | Laws of Logic | VTU BCS405A Q1(b)
🔗    • Prove [¬P∧(¬Q∧R)]∨[(Q∧R)∨(P∧R)] ≡ R | Laws...  

8️⃣ Direct Proof: Square of an Odd Integer is Always Odd | VTU BCS405A Discrete Mathematics
🔗    • Direct Proof: Square of an Odd Integer is ...  

9️⃣ Direct Proof: Sum of Odd Numbers is Even & Product is Odd | VTU BCS405A Logic
🔗    • Direct Proof: Sum of Odd Numbers is Even &...  

🔟 Direct, Indirect & Contradiction Proof Explained | Odd + 9 = Even | VTU Logic Module 1
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1️⃣1️⃣ Quantifiers in Logic | Symbolic Representation Using ∀ and ∃ | Model QP 1 – Q1(c)
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1️⃣2️⃣ Nested Quantifiers & Truth Value | Logic & Use of Quantifiers | VTU BCS405A Q2(c)
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1️⃣3️⃣ Nested Quantifiers | BCS405A Module 1 Q2(c) | VTU Discrete Maths | Truth Values
🔗    • Nested Quantifiers | BCS405A Module 1 Q2(c...  

📂 📺 Playlists by Module:

👉 BCS405A Module 1 – Fundamentals of Logic | VTU Dec 2024, June 2024, Model Papers Solved
🔗    • BCS405A Module 1 – Fundamentals of Logic |...  

▶️ Module 1 – Fundamentals of Logic
🔗    • BCS405A Module 1 ( Fundamentals of Logic )  

▶️ Module 2 – Sets, Relations, Functions
🔗    • BCS405A Module 2( Discrete Mathematics )  

▶️ Module 3 – Graph Theory & Trees
🔗    • Relations and Functions | BCS405A Module 3...  

▶️ Module 4 – Recurrence, Permutations, Inclusion-Exclusion
🔗    • BCS405A Module 4  ( The principle of exclu...  




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