Exploring Group Properties of a Real Number Set

In summary, we are given a set S of real numbers except -1 and an operation * defined as a + b + ab. We are asked to find the values of 3 * 5 and -2 * 6, prove that S with * is a group, and find a solution for the equation 2 * x * 3 = 7 in S. To prove that S with * is a group, we must show that it satisfies the group axioms of closure, associativity, identity, and inverse element. To solve the equation 2 * x * 3 = 7 in S, we can write it out and rearrange it to find a solution.
  • #1
duki
264
0

Homework Statement



Let S be the set of all real numbers except -1.
Define * on S by:

a * b = a + b + ab

Homework Equations



i) find 3 * 5
ii) find -2 * 6
iii) show that S with the operation * is a group
iv) find a solution of the quation 2 * x * 3 = 7 in S

The Attempt at a Solution



Ok, I have no clue on these. Here is what I've attempted:

i) 28
ii) -8
iii) ?
iv) ?

Could someone give me a hand? :)
 
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  • #2
Need to show the set & * satisfy the group axioms
see http://en.wikipedia.org/wiki/Group_(mathematics)"

they are:
1. closure
2. associativity
3. identity
4. inverse element

so for exmpale to show closure, this means for evry 2 elemnt a,b in G, a*b is also in G, so try to show no 2 elements a,b in G can have a*b = -1, as this is the only number not in G

associativity try writing it out & rearranging, then try and find equations for the last 2 given an element

for question iv) try writing it out
 
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  • #3
First (i) is not 28. Since that is just simple arithmetic and you got (b) right, I assume that is a typo.
 

Related to Exploring Group Properties of a Real Number Set

1. What is a real number set?

A real number set is a collection of numbers that includes both rational and irrational numbers. It is represented by the symbol ℝ.

2. What are the group properties of a real number set?

The group properties of a real number set are closure, associativity, identity, and inverse. These properties state that when two real numbers are added, subtracted, multiplied, or divided, the result will always be a real number. Additionally, the order of operations does not affect the final result, and every real number has an additive and multiplicative inverse.

3. How do we explore group properties of a real number set?

To explore the group properties of a real number set, we can use various mathematical operations such as addition, subtraction, multiplication, and division. We can also apply different sets of real numbers and observe how the properties hold true.

4. Why is it important to understand group properties of a real number set?

Understanding the group properties of a real number set is crucial in various areas of mathematics, including algebra, calculus, and geometry. These properties help us manipulate and simplify real number expressions, which is essential in solving equations and proving mathematical theorems.

5. What are some real-life applications of group properties of a real number set?

Group properties of a real number set have practical applications in fields such as physics, engineering, and economics. For example, in physics, these properties are used to calculate the velocity, acceleration, and force of an object. In economics, they are used to analyze financial data and make predictions.

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