Proof Techniques

$$1 + 3 + 5 + \cdots + (2N-1) = N^2$$ --- When N = 1, the first odd integer is 1. - LHS = 1 - RHS = 1² = 1 Since LHS = RHS, the statement holds for N = 1. ✓ --- Assume the statement is true for N = k, i.e., assume: $$1 + 3 + 5 + \cdots + (2k-1) = k^2$$ --- ...

$$1^3 + 2^3 + 3^3 + \cdots + n^3 = \left[\frac{n(n+1)}{2}\right]^2$$ This is a perfect square for all positive integers $n$. --- LHS: $1^3 = 1$ RHS: $\left[\frac{1(1+1)}{2}\right]^2 = \left[\frac{1 \cdot 2}{2}\right]^2 = [1]^2 = 1$ Since LHS $=$ RHS $= 1$, ...

Claim: The sum of the first n odd positive integers is n². The first n odd positive integers are: 1, 3, 5, 7, ..., (2n - 1). So we want to prove: $$1 + 3 + 5 + \cdots + (2n-1) = n^2$$ --- Left-hand side (LHS): The first odd positive integer is 1. Right-hand...

$$P(n): 1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}$$ for all positive integers $n \geq 1$. --- Left Hand Side (LHS): $$1$$ Right Hand Side (RHS): $$\frac{1(1+1)}{2} = \frac{1 \times 2}{2} = 1$$ Since LHS = RHS = 1, the statement P(1) is true. --- Assume that...

--- Mathematical Induction is a proof technique used to establish that a statement P(n) is true for all positive integers n. It operates on a simple but powerful principle: if a statement holds for a starting value, and whenever it holds for some value k it...

Power(a, n)
{
if (n = 0) return 1;
else return a * Power(a, n-1);
}

Claim: For all integers $n \geq 0$, Power(a, n) correctly returns $a^n$. --- When $n = 0$, the algorithm executes the branch: $$\text{return } 1$$ By definition of exponentiation: $$a^0 = 1 \quad \text{for any } a$$ The algorithm returns $1$, which equals $...

Before writing the proof, we state the formal definitions: - An integer $n$ is even if $n = 2k$ for some integer $k$. - An integer $n$ is odd if $n = 2k + 1$ for some integer $k$. --- If $a$ is an odd integer and $b$ is an even integer, then $a + b$ is odd....

An integer $n$ is even if there exists an integer $k$ such that: $$n = 2k$$ --- Statement: If $a$ and $b$ are even integers, then $a + b$ is also an even integer. --- Assume that $a$ and $b$ are two even integers. Step 1: By the definition of even numbers, ...

--- A direct proof is a method in which we assume the hypothesis p is true and then use logical steps, definitions, axioms, and previously proven theorems to directly establish that the conclusion q is true. Structure: We move in a straight forward path fro...

If $n$ is an even number, then $n^2$ is even. --- - An integer $n$ is even if there exists an integer $k$ such that $n = 2k$. - An integer $m$ is even if there exists an integer $j$ such that $m = 2j$. --- Given: $n$ is an even integer. To Prove: $n^2$ is e...

A direct proof is a method of proving a statement of the form "If P, then Q" by assuming P is true and using logical steps, definitions, axioms, and previously proven theorems to show that Q must also be true. It proceeds in a straightforward, linear chain ...

A recursive definition (also called an inductive definition) defines an object in terms of simpler versions of itself. It consists of two parts: 1. Base Case (Basis Step): Defines the simplest instance(s) of the object explicitly. 2. Recursive/Inductive Ste...

- A rational number is any number that can be expressed as $\frac{p}{q}$, where $p, q \in \mathbb{Z}$ and $q \neq 0$, with $\gcd(p, q) = 1$. - An irrational number is a number that cannot be expressed in such a form. --- Assume, for the sake of contradictio...

Rules of inference are valid argument forms used to derive conclusions from premises in logical reasoning. --- $$p$$ $$p \rightarrow q$$ $$\therefore q$$ Meaning: If $p$ is true, and $p \rightarrow q$ is true, then $q$ must be true. Example: "It is raining....