Proofs

Author: SolveSheep

An introduction to the rigorous proof world, where the result must be formal and strict, but with easy things to proof, such us basic properties or maybe mos complicated things




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1.0 Algebra
Commutativity of Set Union

For any two sets \( A \) and \( B \), prove that \( A \cup B = B \cup A \).
1.0 Algebra Number Theory
Difference of Squares of Consecutive Integers

Prove that the difference between the squares of any two consecutive integers is always an odd integer.
1.0 Geometry
Equality of Vertical Angles

Prove that when two straight lines intersect, the vertical angles formed are equal in measure.
1.0 Algebra Inequalities
Non-negativity of a Real Square

For any real number \( x \), prove that \( x^2 \geq 0 \).
1.0 Algebra Number Theory
Product of Two Even Integers

Prove that the product of any two even integers is an even integer.
1.0 Algebra Number Theory
Product of an Even and an Integer

Prove that the product of an even integer and any integer is always an even integer.
1.0 Algebra Number Theory
Sum of Two Odd Integers

Prove that the sum of two odd integers is always an even integer.
1.5 Algebra Inequalities
Absolute Value Product Property

For any real numbers \( a \) and \( b \), prove that \( |a \cdot b| = |a| \cdot |b| \).
1.5 Algebra
Associativity of Set Intersection

For any three sets \( A, B, C \), prove that \( (A \cap B) \cap C = A \cap (B \cap C) \).
1.5 Algebra Number Theory
Closure of Rationals Under Addition

Prove that the sum of any two rational numbers is always a rational number.
1.5 Geometry
Exterior Angle Theorem for Triangles

Given a triangle \( \triangle ABC \), with \( D \) a point on the line containing \( BC \) such that \( C \) is between \( B \) and \( D \). Prove that the measure of the exterior angle \( \angle ACD \) is equal to the sum of the measures of the two opposite interior angles, \( \angle BAC + \angle ABC \).
1.5 Algebra
Intersection with a Superset

For any two sets \( A \) and \( B \), prove that if \( A \subseteq B \), then \( A \cap B = A \).
1.5 Algebra Number Theory
Parity of Square Roots

Prove that if \( x \) is an integer such that \( x^2 \) is an even integer, then \( x \) must also be an even integer.
1.5 Algebra
Set Difference and Union Identity

For any two sets \( A \) and \( B \), prove that \( (A \setminus B) \cup B = A \cup B \).
1.5 Number Theory Combinatorics
Simple Pigeonhole Principle Application

Prove that among any group of 13 people, at least two people must have been born in the same month.
1.5 Geometry
Sum of Interior Angles of a Quadrilateral

Prove that the sum of the measures of the interior angles of any convex quadrilateral is \( 360^\circ \).
1.5 Algebra Number Theory
Transitivity of Divisibility

Let \( a, b, c \) be integers. Prove that if \( a \) divides \( b \) and \( b \) divides \( c \), then \( a \) divides \( c \).
1.5 Algebra
Union with a Superset

For any two sets \( A \) and \( B \), prove that if \( A \subseteq B \), then \( A \cup B = B \).
2.0 Algebra Number Theory
Sum of Rational and Irrational

Prove that the sum of a rational number and an irrational number is always an irrational number.
2.0 Algebra
Uniqueness of Additive Identity

Prove that the additive identity in the set of real numbers (i.e., the number 0 such that \( a+0=a \) for any real \( a \)) is unique.