# Describe a Mathematical System

Properties of Mathematical System

- 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). - 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. - 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 - 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) - 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 - 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.

- 9 + 0 = 9
- 6 x 9 = 9 x 6
- m ( n + o ) = mn + mo
- 1 + (-1) = 0
- d + ( e + f ) = (d + e ) + f
- 8 x 1/8 = 1
- 3 (2n – 1) = 6n – 3
- 11 + 2x = 2x + 11
- 12 x 1 = 12
- (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:**

- 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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