Mathematical Proof Techniques Overview

A method of proving a statement by assuming its negation and finding a contradiction.

The opposite statement of a given statement, denoted as ¬P.

A type of proof that shows if P implies Q (P ⇒ Q).

Situations where an assumption leads to an impossible outcome.

Following a logical sequence to reach a contradiction or conclusion.

An integer of the form 2k + 1, where k is an integer.

An integer divisible by 2, typically of the form 2k.

The result of a proof showing the original statement is true.

When an even integer is squared, the result is always even.

A situation where two statements cannot both be true at the same time.

A number that can be expressed as the quotient of two integers, with a non-zero denominator.

A number that cannot be expressed as a simple fraction, with non-repeating, non-terminating decimals.

An equation involving logarithms, often relating exponents and their bases.

A mathematical statement that establishes a relationship between two expressions using inequality (≤, <, >, ≥).

To show P is true if and only if Q, prove both cases: P ⇒ Q, Q ⇒ P.

This case shows that if statement P is true, then statement Q must also be true.

This case shows that if statement Q is true, then statement P must also be true.

An example that disproves a universal statement by showing it false.

To show that a statement holds for all numbers is false using a counterexample or contradiction.

A method showing that a statement is false by evaluating different scenarios.

A straightforward method to establish the truth of a statement without contradictions or counterexamples.

A function where the value never decreases as input increases.

A function where the value never increases as input increases.

Changing the direction of an inequality when multiplying/dividing by a negative number.

When multiplying/dividing by a positive number, the sign of the inequality remains unchanged.

When the operation matches a non-monotonic function, additional details are necessary to assess the inequality.

Using the tangent function, which can lead to discontinuities in inequalities.

The inverse of a number, which can alter the relationship of sides in inequalities.

The condition where two expressions are equivalent in value.

Prove that 𝑎² - 𝑏²𝑥² - 𝑦² ≤ 𝑎𝑥 - 𝑏𝑦.

If 𝑎, 𝑏 > 0, prove that 𝑎/𝑏 + 𝑏/𝑎 ≥ 2.

Prove that 𝑎² + 𝑏² + 𝑐² + 𝑑² + 𝑒² ≥ 𝑎𝑏 + 𝑐 + 𝑑 + 𝑒.

By substituting 𝑎 = 2𝑥, 𝑏 = 𝑦, prove that 𝑥 + 𝑦 ≥ 9 for 𝑥, 𝑦 > 0.

By substituting 𝑎 = 𝑥, 𝑏 = 𝑦, prove that 𝑥 + 𝑦 ≥ 8.

A method of proving a statement by showing that if the conclusion is false, the hypothesis must also be false.

A relationship where two statements can be shown to imply each other, making them logically equivalent.

The act of demonstrating that a certain statement or hypothesis is false.

An inequality stating that for any non-negative set of numbers, the arithmetic mean is greater than or equal to the geometric mean.

A number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit.

The standard form of a complex number, represented as a + bi.

A way to express complex numbers in terms of their modulus and argument.

A formula that relates complex numbers and trigonometry, expressing (cos θ + i sin θ)^n.

A method of proving statements for all natural numbers by establishing a base case and an inductive step.

A technique used to integrate products of functions by applying the formula ∫u dv = uv - ∫v du.

Vectors that have three components, typically represented as (x, y, z).

The motion of an object that is thrown into the air and influenced by gravitational force.

A type of periodic motion where the restoring force is directly proportional to the displacement.