Describe a Mathematical System

Properties of Mathematical System

  1. Closure. S is closed under the operation (). If for any two elements a, b
    in S, the element (a)(b) is also in S. For example, the naturals are closed
    under addition since adding any two naturals always results in another,
    bigger, natural (e.g. 2 + 6 = 8).
  2. Identity Element. S has an identity element i. If for any element a in S,
    a(i) = (i ) a = a. Example: a. 3(1) = (1)3 = 3, 1 is the identity element.
    b. 3 + 0 = 3, 0 is the identity element.
  3. Inverse Element. If an element b in S is such that a(b) = (b)a = i, then this
    element is called the inverse of a and (a, b) is called an inverse pair.
    Example: a. 3 + (-3) = 0 b. 3(1
    3
    ) = 1
  4. Associative Property. If for any three elements a, b, c in S, (a)(b) ∗ c = a
    ∗ (b ∗ c), the operation ∗ is said to be associative.
    Example: a. (2 + 3) + 5 = 2 + (3 + 5) b. (2 x 3) 5 = 2 (3 x 5)
  5. Commutative Property. If for any two elements a, b in S, a ∗ b = b ∗ a, the
    operation ∗ is said to be commutative.
    Example: a. 3 + 4 = 4 + 3 b. 3 (4) = (4) 3
  6. Distributive Property. The sum of two numbers times a third number is
    equal to the sum of each addend time the third number.

Exercise 1

Direction: Write TRUE if the statement is correct and FALSE if not. 

_______________ 1. Mathematical system is a structure formed from    undefined objects only. 

_______________ 2. Theorems are assertions about the properties of the     universe and rules for creating and justifying more     assertions. 

_______________ 3. A mathematical system consists of a set elements.

_______________ 4. Descriptions explain the meaning of concepts that relate     to something. 

                                         _______________ 5. If an element b in S is such that a  b = b  a = i, then     this element is called the inverse of a and (a, b) is     called an reverse pair.

    _______________ 6. x  y = y  x is an example of commutative property.

           _______________ 7. The additive inverse of 7 is -7.

             _______________ 8. Commutative property is also true to subtraction. 

   _______________ 9. For any three elements x, y, z in S, (x  y)  z = x  (y   

   z), the operation  is said to be associative.

           _______________ 10. 5 + 0 = 5. 0 is the identity element.

Exercise 1 Answers:

1. False

2. False

3. True

4. False

5. False

6. True

7. True

8. False

9. True

10. True

Exercise 2

Directions:State the property of mathematical system shown below.

  1. 9 + 0 = 9
  2. 6 x 9 = 9 x 6
  3. m ( n + o ) = mn + mo
  4. 1 + (-1) = 0
  5. d + ( e + f ) = (d + e ) + f
  6. 8 x 1/8 = 1
  7. 3 (2n – 1) = 6n – 3
  8. 11 + 2x = 2x + 11
  9. 12 x 1 = 12
  10. (3x + y) + 5 = 3x + (y + 5)

Exercise 2 Answers

1.Identity

2. Commutative

3. Distributive

4. Inverse

5. Associative

6. Inverse

7. Distributive

8. Commutative

9. Identity

10. Associative

       V.       Reflection:

  1. What have you learned in the lesson?

A structure formed from one or more sets of undefined objects, various

concepts which may or may not be defined, and a set of axioms relating these objects and concepts is called mathematical system.

  • How can you apply it to real- life situation/s?

I can apply it to real-life situations like computations and other mathematical operations.

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