Direct Proof

Assuming hypothesis p to be true and proving the conclusion q to be true, in order to prove that p ⇒ q is true

Proof by Contraposition

Showing the logical equivalence of a conditional proposition to its contraposition

p → q ≡ ¬q → ¬p

Examples

Check useful laws if you don't recall these laws

Existence Proof

Proving the proposition by providing a True example

Examples

1. For some real numbers x, y prove x^2 > y^2 if x > y

Equation is true for x = 2 and y = -1.

Therefore, the proposition is True for some real numbers.

2. For some real number a, prove that (a+1)^2 >= a^2

Equation is true for a = 3.

Therefore, the proposition is True for some real numbers.

Proof by Counter-Example

Proving the proposition by providing a False example

Examples

1. For all real numbers x, y prove x^2 > y^2 if x > y

Equation is false for x = 0 and y = -1.

Therefore, the proposition is not True for all real numbers.

2. For all real number a, prove that (a+1)^2 >= a^2

Equation is false for a = -1.

Therefore, the proposition is not True for all real numbers.