Algebraic Properties, Number Sets, Closure, and Linear Proofs

Properties of Real Number Operations and Identities

• Commutative Property of Addition:

  • Formal Definition: For any real numbers $a$ and $b$, changing the order of the addends does not alter the sum: $a + b = b + a$.
  • Numeric Example: $7 + 12 = 12 + 7$.
  • Algebraic Example: $14 + x = x + 14$.

• Commutative Property of Multiplication:

  • Formal Definition: For any real numbers $a$ and $b$, changing the order of the factors does not alter the product: $a \times b = b \times a$ (or $a b = b a$).
  • Numeric Example: $4(5) = 5(4)$.
  • Algebraic Example: $y(4) = 4y$.

• Associative Property of Addition:

  • Formal Definition: For any real numbers $a$, $b$, and $c$, the grouping of addends does not affect the resulting sum: $(a + b) + c = a + (b + c)$.
  • Numeric Example: $(6 + 3) + 4 = 6 + (3 + 4)$.

• Associative Property of Multiplication:

  • Formal Definition: For any real numbers $a$, $b$, and $c$, the grouping of factors does not affect the resulting product: $(a b)c = a(b c)$.
  • Numeric Example: $(4(3))(5) = 4(3(5))$.
  • Algebraic Example: $(6(2))y = 6(2y)$.

• Distributive Property of Multiplication over Addition:

  • Formal Definition: Multiplication distributes over addition from the left and from the right: $a(b + c) = a b + a c$ and $(b + c)a = b a + c a$.
  • Numeric Example: $3(4 + 2) = 3(4) + 3(2)$.
  • Algebraic Example with Subtraction: $8(y - 3) = 8y - 8(3)$.

• Additive Identity Property:

  • Formal Definition: The sum of any real number $a$ and zero equals the original number: $a + 0 = 0 + a = a$.
  • Numeric Example: $4 + 0 = 0 + 4 = 4$.
  • Algebraic Example: $x + 0 = 0 + x = x$.

• Multiplicative Identity Property:

  • Formal Definition: The product of any real number $a$ and one equals the original number: $1(a) = a(1) = a$.
  • Numeric Example: $1(13) = 13(1) = 13$.
  • Algebraic Example: $1(y) = y(1) = y$.

• Additive Inverse Property:

  • Formal Definition: For every real number $a$, there exists an opposite $-a$ such that their sum is the additive identity, zero: $a + (-a) = 0$.
  • Numeric Integer Example: $-4 + 4 = 0$.
  • Radical Example: $\sqrt{2} + (-\sqrt{2}) = 0$.
  • Compound Algebraic Example: $(a + b) + (-(a + b)) = 0$.

• Multiplicative Inverse Property:

  • Formal Definition: For every nonzero real number $a$, there exists a reciprocal $\frac{1}{a}$ such that their product is the multiplicative identity, one: $a\left(\frac{1}{a}\right) = 1$.
  • Numeric Example: $2\left(\frac{1}{2}\right) = 1$.
  • Algebraic Example: $y\left(\frac{1}{y}\right) = 1$.

• Multiplication Property of Zero:

  • Formal Definition: The product of any real number $a$ and zero is zero: $0(a) = a(0) = 0$.
  • Numeric Example: $0(9) = 9(0) = 0$.
  • Algebraic Example: $0(y) = y(0) = 0$.

Classification and Hierarchy of Real Number Subsets

• Standard Subsets of the Real Number System:

  • Natural Numbers ($\mathbb{N}$): Counting numbers starting at one: $\{1, 2, 3, 4, \dots\}$.
  • Whole Numbers ($\mathbb{W}$): Non-negative counting numbers including zero: $\{0, 1, 2, 3, \dots\}$.
  • Integers ($\mathbb{Z}$): Positive and negative counting numbers along with zero: $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$.
  • Rational Numbers ($\mathbb{Q}$): Any number expressible as the quotient of two integers $\frac{a}{b}$ where $b \neq 0$, exhibiting terminating or repeating decimal expansions.
  • Irrational Numbers ($\mathbb{I}$): Numbers that cannot be expressed as a simple fraction, exhibiting non-terminating and non-repeating decimal expansions.
  • Real Numbers ($\mathbb{R}$): The union of all rational and irrational numbers.

• Classifications of Specific Numbers:

  • $0.232232223\dots$: Non-terminating, non-repeating pattern; classified as Real, Irrational.
  • $\sqrt{3}$: Non-terminating, non-repeating square root of a non-perfect square; classified as Real, Irrational.
  • $-3\sqrt{16}$: Evaluates directly to $-3(4) = -12$; classified as Real, Rational, Integer.
  • $0.121212\dots$: Repeating decimal expansion equal to $\frac{12}{99} = \frac{4}{33}$; classified as Real, Rational.
  • $-3.75$: Terminating decimal expansion equal to $-\frac{15}{4}$; classified as Real, Rational.
  • $\sqrt{4}$: Evaluates directly to $2$; classified as Real, Rational, Integer, Whole, Natural.
  • $7$: Positive non-fractional integer; classified as Real, Rational, Integer, Whole, Natural.

Mathematical Closure across Sets and Operations

• Concept of Closure:

  • A set is defined as closed under an operation if performing that operation on any elements of the set always produces an element that is also a member of that set.

• Closure Properties of Fundamental Number Sets:

  • Addition: Closed for Integers, Natural numbers, Whole numbers, Rational numbers, and Real numbers.
  • Subtraction: Closed for Integers, Rational numbers, and Real numbers. Natural numbers and Whole numbers are NOT closed under subtraction (e.g., $2 - 5 = -3$, which is neither whole nor natural).
  • Multiplication: Closed for Natural numbers, Whole numbers, Integers, Rational numbers, and Real numbers.
  • Division: Closed for Rational numbers and Real numbers (excluding division by zero). Integers, Whole numbers, and Natural numbers are NOT closed under division (e.g., $1 \div 2 = 0.5$, which is not an integer).

• Closure Analysis of Specific Sets:

  • Positive Rational Numbers: Not closed under Subtraction. Subtracting a larger positive rational from a smaller positive rational yields a negative value (e.g., $2 - 5 = -3$), and subtracting equal values yields zero ($2 - 2 = 0$), neither of which belongs to the set of positive rationals.
  • The Set of Powers of Two ($\{1, 2, 4, 8, 16, \dots\}$ or $\{2^n \mid n \in \mathbb{W}\}$):
    • Not closed under Addition (e.g., $1 + 2 = 3$, and $3 \notin \{1, 2, 4, 8, 16, \dots\}$).
    • Not closed under Subtraction (e.g., $1 - 2 = -1$, and $-1 \notin \{1, 2, 4, 8, 16, \dots\}$).
    • Not closed under Division (e.g., $1 \div 2 = 0.5$, and $0.5 \notin \{1, 2, 4, 8, 16, \dots\}$).

• Operations on Distinct Irrational Numbers:

  • Given distinct irrational numbers $a$, $b$, and $c$:
    • Quotients ($a \div b$) can compute a rational number (e.g., $\sqrt{8} \div \sqrt{2} = 2$).
    • Products ($a \times b$) can compute a rational number (e.g., $\sqrt{2} \times \sqrt{8} = 4$).
    • Sums of additive inverses ($a + (-a)$) compute $0$, which is rational, but require identical absolute terms rather than distinct irrationals.

Linear Inequalities: Algebraic Solution, Graphing, and Interval Notation

• Linear Inequality Problem:

  • Given inequality statement:         $-4p + 28 \ge -8$

• Step-by-Step Solution Procedure:

  • Subtract $28$ from both sides of the inequality using the Subtraction Property of Inequality:         $-4p + 28 - 28 \ge -8 - 28$$-4p \ge -36$
  • Divide both sides by $-4$ and invert the direction of the inequality sign using the Division Property of Inequality:         $\frac{-4p}{-4} \le \frac{-36}{-4}$$p \le 9$

• Graphical Representation on a Number Line:

  • The boundary point occurs at $9$.
  • A closed circle or bracket is placed at $9$ to indicate inclusion ($\le$).
  • The shading extends continuously to the left towards negative infinity across points such as $8$, $7$, and $6$.

• Interval Notation:

  • The complete solution set is expressed in interval notation as:         $(-\infty, 9]$

Step-by-Step Algebraic Equation Proofs and Axiomatic Justifications

• Proof 1: Solution to $5x - 3 - 2x = 12$:

  • Step 1:         $5x - 3 - 2x = 12$         Justification: Given.
  • Step 2:         $3x - 3 = 12$         Justification: Distributive Property (combining like terms: $(5 - 2)x = 3x$).
  • Step 3:         $3x - 3 + 3 = 12 + 3$         Justification: Addition Property of Equality.
  • Step 4:         $3x = 15$         Justification: Additive Inverse Property ($-3 + 3 = 0$) and Additive Identity Property ($3x + 0 = 3x$).
  • Step 5:         $\frac{3x}{3} = \frac{15}{3}$         Justification: Division Property of Equality (or Multiplication Property of Equality multiplying by $\frac{1}{3}$).
  • Step 6:         $x = 5$         Justification: Multiplicative Inverse Property ($\frac{3}{3} = 1$) and Multiplicative Identity Property ($1x = x$).

• Proof 2: Solution to $2x - 4 + 3x - 9 = 6x - 13$:

  • Step 1:         $2x - 4 + 3x - 9 = 6x - 13$         Justification: Given.
  • Step 2:         $5x - 13 = 6x - 13$         Justification: Distributive Property (combining like terms: $(2 + 3)x = 5x$ and $-4 - 9 = -13$).
  • Step 3:         $5x - 13 - 5x + 13 = 6x - 13 - 5x + 13$         Justification: Addition Property of Equality and Subtraction Property of Equality.
  • Step 4:         $0 = x$         Justification: Additive Inverse Property and Additive Identity Property (resulting in $x = 0$).

• Proof 3: Solution to $4(x - 5) - 2x = 3x - 8$:

  • Step 1:         $4(x - 5) - 2x = 3x - 8$         Justification: Given.
  • Step 2:         $4x - 20 - 2x = 3x - 8$         Justification: Distributive Property of Multiplication over Addition.
  • Step 3:         $2x - 20 = 3x - 8$         Justification: Distributive Property (combining like terms: $(4 - 2)x = 2x$).
  • Step 4:         $2x - 20 - 2x = 3x - 8 - 2x$         Justification: Subtraction Property of Equality.
  • Step 5:         $-20 = x - 8$         Justification: Additive Inverse Property and Additive Identity Property.
  • Step 6:         $-20 + 8 = x - 8 + 8$         Justification: Addition Property of Equality.
  • Step 7:         $-12 = x$         Justification: Additive Inverse Property and Additive Identity Property (yielding $x = -12$).