Common Mathematical Expressions in Proofs

Dec 13, 2025 9 minutes to read cheat sheet

To construct effective proofs, it’s important to translate statements into precise mathematical language. Below is a summary of common expressions and definitions used in proofs, along with their logical formulations.

Even and Odd Integers

Even Integer:
- Definition: An integer \( n \) is even if there exists an integer \( k \) such that \( n = 2k \).
- Logical Formulation: \( \exists k \in \mathbb{Z},\ n = 2k \)
- Example 1:
  - \( n = 8 \)
  - \( 8 = 2 \times 4 \), so \( 8 \) is even.
- Example 2:
  - \( n = -6 \)
  - \( -6 = 2 \times (-3) \), so \( -6 \) is even.
- Example 3:
  - \( n = 0 \)
  - \( 0 = 2 \times 0 \), so \( 0 \) is even.
- Pronunciation: even integer; n equals two k
- Alternative Terms: Divisible by 2
Odd Integer:
- Definition: An integer \( n \) is odd if there exists an integer \( k \) such that \( n = 2k + 1 \).
- Logical Formulation: \( \exists k \in \mathbb{Z},\ n = 2k + 1 \)
- Example 1:
  - \( n = 7 \)
  - \( 7 = 2 \times 3 + 1 \), so \( 7 \) is odd.
- Example 2:
  - \( n = -9 \)
  - \( -9 = 2 \times (-5) + 1 \), so \( -9 \) is odd.
- Example 3:
  - \( n = 1 \)
  - \( 1 = 2 \times 0 + 1 \), so \( 1 \) is odd.
- Pronunciation: odd integer; n equals two k plus one

Divisibility

Definition: An integer \( a \) divides an integer \( b \) (written \( a \mid b \)) if there exists an integer \( k \) such that \( b = ak \).
Logical Formulation: \( a \mid b \iff \exists k \in \mathbb{Z},\ b = ak \)
Example 1:
- \( 5 \mid 20 \) because \( 20 = 5 \times 4 \).
Example 2:
- \( -3 \mid 9 \) because \( 9 = (-3) \times (-3) \).
Example 3:
- \( 4 \mid 0 \) because \( 0 = 4 \times 0 \).
Pronunciation: a divides b; there exists an integer k such that b equals a times k
Alternative Terms: \( b \) is a multiple of \( a \); \( a \) is a factor of \( b \)

Prime and Composite Numbers

Prime Number:
- Definition: An integer \( p > 1 \) is prime if its only positive divisors are \( 1 \) and \( p \).
- Logical Formulation: \( p > 1 \land \forall d \in \mathbb{N},\ (d \mid p \implies d = 1 \lor d = p) \)
- Example 1:
  - \( p = 2 \) is prime.
- Example 2:
  - \( p = 13 \) is prime.
- Example 3:
  - \( p = 17 \) is prime.
- Pronunciation: prime number
Composite Number:
- Definition: An integer \( n > 1 \) is composite if it is not prime.
- Logical Formulation: \( n > 1 \land \exists d \in \mathbb{N},\ 1 < d < n,\ d \mid n \)
- Example 1:
  - \( n = 4 \) because \( 2 \mid 4 \).
- Example 2:
  - \( n = 15 \) because \( 3 \mid 15 \) and \( 5 \mid 15 \).
- Example 3:
  - \( n = 21 \) because \( 3 \mid 21 \) and \( 7 \mid 21 \).
- Pronunciation: composite number

Rational and Irrational Numbers

Rational Number:
- Definition: A real number \( r \) is rational if there exist integers \( a \) and \( b \neq 0 \) such that \( r = \frac{a}{b} \).
- Logical Formulation: \( \exists a, b \in \mathbb{Z},\ b \neq 0,\ r = \frac{a}{b} \)
- Example 1:
  - \( r = 0.5 = \frac{1}{2} \).
- Example 2:
  - \( r = -3 = \frac{-3}{1} \).
- Example 3:
  - \( r = 0 = \frac{0}{1} \).
- Pronunciation: rational number; r equals a over b
Irrational Number:
- Definition: A real number that is not rational.
- Logical Formulation: \( \lnot (\exists a, b \in \mathbb{Z},\ b \neq 0,\ r = \frac{a}{b}) \)
- Example 1:
  - \( r = \pi \).
- Example 2:
  - \( r = e \) (Euler’s number).
- Example 3:
  - \( r = \sqrt{3} \).
- Pronunciation: irrational number

Congruence Modulo \( n \)

Definition: Integers \( a \) and \( b \) are congruent modulo \( n \) (written \( a \equiv b \mod n \)) if \( n \mid (a - b) \).
Logical Formulation: \( a \equiv b \mod n \iff n \mid (a - b) \)
Example 1:
- \( 14 \equiv 2 \mod 12 \) because \( 12 \mid (14 - 2) = 12 \).
Example 2:
- \( -1 \equiv 4 \mod 5 \) because \( 5 \mid (-1 - 4) = -5 \).
Example 3:
- \( 100 \equiv 0 \mod 10 \) because \( 10 \mid (100 - 0) = 100 \).
Pronunciation: a is congruent to b modulo n

Floor and Ceiling Functions

Floor Function:
- Definition: \( \lfloor x \rfloor \) is the greatest integer less than or equal to \( x \).
- Logical Formulation: \( \lfloor x \rfloor = \text{max} \{ n \in \mathbb{Z} \mid n \leq x \} \)
- Example 1:
  - \( \lfloor 2.9 \rfloor = 2 \).
- Example 2:
  - \( \lfloor -1.2 \rfloor = -2 \).
- Example 3:
  - \( \lfloor 5 \rfloor = 5 \).
- Pronunciation: floor of x
Ceiling Function:
- Definition: \( \lceil x \rceil \) is the least integer greater than or equal to \( x \).
- Logical Formulation: \( \lceil x \rceil = \text{min} \{ n \in \mathbb{Z} \mid n \geq x \} \)
- Example 1:
  - \( \lceil 3.1 \rceil = 4 \).
- Example 2:
  - \( \lceil -0.7 \rceil = 0 \).
- Example 3:
  - \( \lceil 6 \rceil = 6 \).
- Pronunciation: ceiling of x

Absolute Value

Definition: The absolute value of \( x \), denoted \( |x| \), is: \[ |x| = \begin{cases} x, & \text{if } x \geq 0 \ -x, & \text{if } x < 0 \end{cases} \]
Logical Formulation:
- \( |x| = x \) if \( x \geq 0 \).
- \( |x| = -x \) if \( x < 0 \).
Example 1:
- \( |7| = 7 \).
Example 2:
- \( |-3| = 3 \).
Example 3:
- \( |0| = 0 \).
Pronunciation: absolute value of x
Alternative Terms: Modulus

Inequalities

Transitivity:
- Definition: If \( a > b \) and \( b > c \), then \( a > c \).
- Logical Formulation: \( (a > b \land b > c) \implies a > c \)
- Example 1:
  - If \( 5 > 3 \) and \( 3 > 1 \), then \( 5 > 1 \).
- Example 2:
  - If \( -2 > -5 \) and \( -5 > -8 \), then \( -2 > -8 \).
- Example 3:
  - If \( x > y \) and \( y > z \), then \( x > z \).
- Pronunciation: transitivity of inequalities
Addition/Subtraction:
- Definition: If \( a > b \), then \( a + c > b + c \).
- Logical Formulation: \( (a > b) \implies (a + c > b + c) \)
- Example 1:
  - \( 4 > 2 \implies 4 + 3 > 2 + 3 \) (i.e., \( 7 > 5 \)).
- Example 2:
  - \( -1 > -3 \implies -1 - 2 > -3 - 2 \) (i.e., \( -3 > -5 \)).
- Example 3:
  - \( x > y \implies x - k > y - k \).
- Pronunciation: addition property of inequalities
Multiplication/Division:
- Definition:
  - If \( a > b \) and \( c > 0 \), then \( ac > bc \).
  - If \( c < 0 \), then \( ac < bc \).
- Logical Formulation:
  - \( (a > b \land c > 0) \implies ac > bc \).
  - \( (a > b \land c < 0) \implies ac < bc \).
- Example 1:
  - \( 3 > 1 \) and \( 2 > 0 \) imply \( 3 \times 2 > 1 \times 2 \) (i.e., \( 6 > 2 \)).
- Example 2:
  - \( -2 > -4 \) and \( -1 < 0 \) imply \( -2 \times -1 < -4 \times -1 \) (i.e., \( 2 < 4 \)).
- Example 3:
  - If \( x > y \) and \( c < 0 \), then \( xc < yc \).
- Pronunciation: multiplication property of inequalities

Basic Algebraic Identities

Distributive Law:
- Definition: \( a(b + c) = ab + ac \)
- Logical Formulation: For all \( a, b, c \in \mathbb{R} \), \( a(b + c) = ab + ac \)
- Example 1:
  - \( 2(3 + 4) = 2 \times 3 + 2 \times 4 \).
- Example 2:
  - \( -1(x - y) = -x + y \).
- Example 3:
  - \( k(m + n) = km + kn \).
- Pronunciation: distributive law
Associative Law:
- Addition:
  - Definition: \( (a + b) + c = a + (b + c) \)
  - Logical Formulation: For all \( a, b, c \in \mathbb{R} \), \( (a + b) + c = a + (b + c) \)
- Multiplication:
  - Definition: \( (ab)c = a(bc) \)
  - Logical Formulation: For all \( a, b, c \in \mathbb{R} \), \( (ab)c = a(bc) \)
- Example 1 (Addition):
  - \( (1 + 2) + 3 = 1 + (2 + 3) \).
- Example 2 (Multiplication):
  - \( (2 \times 3) \times 4 = 2 \times (3 \times 4) \).
- Example 3 (Variables):
  - \( (x + y) + z = x + (y + z) \).
- Pronunciation: associative law
Commutative Law:
- Addition:
  - Definition: \( a + b = b + a \)
  - Logical Formulation: For all \( a, b \in \mathbb{R} \), \( a + b = b + a \)
- Multiplication:
  - Definition: \( ab = ba \)
  - Logical Formulation: For all \( a, b \in \mathbb{R} \), \( ab = ba \)
- Example 1 (Addition):
  - \( 5 + 7 = 7 + 5 \).
- Example 2 (Multiplication):
  - \( 4 \times 9 = 9 \times 4 \).
- Example 3 (Variables):
  - \( x \times y = y \times x \).
- Pronunciation: commutative law

Techniques for Proofs

Expressing Statements Precisely:
- Example 1:
  - Verbal: “The square of an odd integer is odd.”
  - Mathematical: If \( n = 2k + 1 \), then \( n^2 = 2m + 1 \) for some \( m \in \mathbb{Z} \).
- Example 2:
  - Verbal: “The sum of two rational numbers is rational.”
  - Mathematical: \( \forall a, b \in \mathbb{Q},\ a + b \in \mathbb{Q} \).
- Example 3:
  - Verbal: “The product of a non-zero rational number and an irrational number is irrational.”
  - Mathematical: \( \forall r \in \mathbb{Q},\ r \neq 0,\ \forall s \in \mathbb{R} \setminus \mathbb{Q},\ rs \in \mathbb{R} \setminus \mathbb{Q} \).
- Pronunciation: expressing statements precisely
Introducing Variables:
- Example 1:
  - Let \( n \in \mathbb{N} \) represent an arbitrary natural number.
- Example 2:
  - Suppose \( x \in \mathbb{R} \) is such that \( x > 0 \).
- Example 3:
  - Let \( \epsilon > 0 \) be given.
- Pronunciation: introducing variables
Manipulating Equations:
- Example 1:
  - Given \( a + b = c \), then \( b = c - a \).
- Example 2:
  - If \( x^2 = y \), then \( x = \pm \sqrt{y} \).
- Example 3:
  - From \( ab = ac \), if \( a \neq 0 \), then \( b = c \).
- Pronunciation: manipulating equations