Mathematics
Grade 9
15 min
Subtract - numbers up to 5
Subtract - numbers up to 5
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1
Introduction & Learning Objectives
Learning Objectives
Define a binary operation on a finite set and test it for the closure property.
Model subtraction using function notation, specifying the domain and calculating the range within the set S = {0, 1, 2, 3, 4, 5}.
Apply modular arithmetic to ensure subtraction operations on a finite set are closed.
Analyze the properties of subtraction, such as the lack of identity and inverse elements in a standard context.
Construct and solve simple polynomial equations whose roots are derived from subtraction within the set S.
Evaluate complex expressions where simple subtraction is embedded within radicals or rational functions.
You've known that 4 - 1 = 3 since first grade. But what if the only numbers that existed were {0, 1, 2, 3, 4, 5}? What would 1 - 4 equal the...
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Key Concepts & Vocabulary
TermDefinitionExample
Finite SetA set containing a specific, countable number of elements. For this lesson, our primary set is S = {0, 1, 2, 3, 4, 5}.The set of possible outcomes when rolling a standard die, {1, 2, 3, 4, 5, 6}, is a finite set.
Binary OperationA rule for combining two elements from a set to produce a third element. Subtraction is a binary operation.Given elements 5 and 2 from our set S, the binary operation of subtraction yields 3: 5 - 2 = 3.
Closure PropertyA set is 'closed' under an operation if performing that operation on any two members of the set always results in an element that is also in the set.The set of integers is closed under subtraction because subtracting any two integers results in another integer. However, the set S = {0, 1, 2, 3, 4, 5} is NOT...
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Core Formulas
Standard Subtraction
c = a - b
The fundamental operation. When analyzing properties like closure, you must check if the result 'c' is an element of the original set.
Function Notation for Subtraction
f_k(x) = x - k
This reframes 'subtracting k' as a function f_k(x). It allows us to use concepts like domain and range to analyze the operation on a given set.
Subtraction with Modulo n
c = (a - b) \pmod{n}
This rule redefines subtraction to force closure on a finite set. The result is the remainder when (a - b) is divided by n. For our set S = {0, 1, 2, 3, 4, 5}, the natural modulus is 6.
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Challenging
Consider the operation a ∗ b = (a - b) mod 5 on the set T = {0, 1, 2, 3, 4}. An element 'z' is a 'zero divisor' if z ≠ 0 and there exists a y ≠ 0 such that z ∗ y = 0 or y ∗ z = 0. Which statement is true?
A.This system has no zero divisors.
B.Every non-zero element is a zero divisor.
C.Only 2 and 3 are zero divisors.
D.The element 1 is a zero divisor.
Challenging
A quadratic polynomial P(x) = x² + bx + c has roots r₁ = (5 - 2) and r₂ = (4 - 3). Find the value of (b - c) mod 6.
A.3
B.1
C.4
D.5
Challenging
Let the operation be defined as a ⊕ b = (k - a - b) mod 6 on the set S = {0, 1, 2, 3, 4, 5}. For what value of k ∈ S does the element 0 act as the identity element for this operation?
A.k = 1
B.k = 5
C.k = 0
D.No value of k works.
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