MMW language of binary operations (1).pptx

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1. ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) MATHEMATICS in the MODERN WORLD INTRODUCTION: INSTRUCTOR: RHEA C. GATON

2. ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) LANGUAGE OF BINARY OPERATION

3. ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) It is an operation that takes two input elements from a set and gives a unique result that also belongs to the same set. A very simple example is the operation of addition. In “3 + 5 = 8,” the operation “+” takes to real numbers 3 and 5 and gives the result 8 which is also a real number. B. LANGUAGE OF BINARY OPERATION

4. ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) A GROUP is a set of elements, with one operation, that satisfies the following properties: ( i ) the set is closed with respect to the operation, (ii) the operation satisfies the associative property, (iii) there is an identity element, and (iv) each element has an inverse.

5. ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) In other words, group is an ordered pair (G, *) where G is a set and * is a binary operation on G satisfying the four properties: Closure Property. If any two elements are combined using the operation, the result must be an element of the set. a * b= c ∈ G, for all a, b, c ∈ G Associative Property (a * b) * c = a * (b * c), for all a, b, c ∈ G. Identity Property. There exists an element e in G, such that for all a ∈ G, A * e = e * a. Inverse Property. For each a ∈ G there is an element a−1 of G, such that a * a−1 = a−1 * a = e.

6. ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) EXAMPLE: Determine whether the set of all non-negative integers under addition is a group. SOLUTION: We will apply the four properties to test the set of all non-negative integers under addition is a group. Step 1 : To test for closure property, we choose any two positive integers, for example 8 + 4 = 12 and 5 + 10 = 15 Notice that the sum of two numbers of a set, the result is always a number of the set. Thus, it is closed. Step 2 : To test for associative property, we choose three positive integers, for example 3 + (2 + 4) = 3 + 6 = 9 (3 + 2) + 4 = 5 + 4 = 9 Thus, it also satisfies the associative property.

7. ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) Step 3: To test for identity property, we chose any positive integer, for example 8 + 0 = 8; 9 + 0 = 9; 15 + 0 = 15. Thus, it also satisfies the identity property. Step 4 : To test for inverse property, we choose any positive integer, for example 4 + (-4) = 0; 10 + (-10) = 0; 23 + (-23) = 0 Note that 𝑎−1 = -a. Thus, it is also satisfying the inverse property. Thus, the set of all non-negative integers under addition is a group, since it satisfies the four properties.

8. ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) Determine whether the set of all rational numbers under multiplication is a group. ACTIVITY: