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Chapter 4: Problem 19

Let \(S\) be the set of all real numbers except \(-1\). Define \(*\) on \(S\) by $$ a \cdot b=a+b+a b \text {. } $$ a. Show that * gives a binary operation on \(S\). b. Show that \((S, *)\) is a group. c. Find the solution of the equation \(2+x * 3=7\) in \(S\).

Short Answer

Expert verified
\((S, *)\) is a group, and the solution for \(2+x * 3=7\) is \(x = \frac{2}{3}\).

Step by step solution

01

Show * as a Binary Operation on S

To demonstrate * is a binary operation on \( S \), we need to show that for all \( a, b \) in \( S \), the operation \( a \cdot b \) results in another element in \( S \).The formula given for the operation is \( a \cdot b = a + b + ab \). Since \( S \) is the set of all real numbers except \(-1\), we observe that for any real numbers \( a, b \) where neither \( a \) nor \( b \) is \(-1\), the result \( a + b + ab \) is also a real number and cannot equal \(-1\). Therefore, \( a \cdot b \) is in \( S \), so \( * \) is a binary operation on \( S \).
02

Check Associativity in (S, *)

For \( * \) to be associative, it must satisfy \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \) for all \( a, b, c \) in \( S \).Compute \( (a \cdot b) \cdot c \):1. Find \( a \cdot b = a + b + ab \).2. Then find \((a \cdot b) \cdot c = (a+b+ab) + c + (a+b+ab)c = a+b+c+ab+ac+bc+abc\).Compute \( a \cdot (b \cdot c) \):1. Find \( b \cdot c = b + c + bc \).2. Then find \( a \cdot (b \cdot c) = a + (b+c+bc) + a(b+c+bc) = a+b+c+ab+ac+abc\).Since both expressions are equal, * is associative.
03

Identify the Identity Element in (S, *)

An identity element \( e \) in \( S \) satisfies \( a \cdot e = a \) and \( e \cdot a = a \) for all \( a \in S \).Set \( a \cdot e = a + e + ae = a \). Solving yields:\[(a+e+ae = a) \Rightarrow e(1+a) = 0 \Rightarrow e = 0\] since \( a eq -1 \).So, 0 is the identity element in \( S \), since \( a \cdot 0 = a \) and \( 0 \cdot a = a \).
04

Find the Inverse Element in (S, *)

An inverse element \( b \) for \( a \) in \( S \) satisfies \( a \cdot b = e \) where \( e = 0 \), the identity.Set \( a + b + ab = 0 \). Solving for \( b \):\[ b = \frac{-a}{1+a} \]Since \( a eq -1 \), \( b \) remains in \( S \). Therefore, the inverse of \( a \) is \( b = \frac{-a}{1+a} \).
05

Solve the Equation \(2+x * 3=7\) in S

Solve for \( x \) in the equation \( 2 + x + 2x + 3 = 7 \).Simplify the equation:\\[ 2 + x + 2x + 3 = 7 \Rightarrow 3x + 5 = 7 \] \[ \Rightarrow 3x = 2 \Rightarrow x = \frac{2}{3} \]Thus, the solution in \( S \) is \( x = \frac{2}{3} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binary Operation
A binary operation is a fundamental concept in group theory that involves combining two elements to produce another element. For a set with a defined binary operation to be valid, the result must stay within the set. In our case, we define a binary operation \(*\) on the set \(S\), which consists of all real numbers except \(-1\). The operation is given as \(a \cdot b = a + b + ab\).
  • For any two elements \(a\) and \(b\) in \(S\), neither being \(-1\), the result \(a + b + ab\) will also be a real number that is not \(-1\).
  • This ensures that the operation \(*\) is closed within \(S\), thereby making it a binary operation on the set \(S\).
Associative Property
The associative property is a critical feature for \(S, *\) being a group, which requires that the grouping of operations does not affect the outcome, or mathematically, \((a \cdot b) \cdot c = a \cdot (b \cdot c)\).
To verify this,
  • Compute \((a \cdot b) \cdot c\): First, calculate \(a \cdot b = a + b + ab\). Then, perform \((a + b + ab) \cdot c = (a+b+ab) + c + (a+b+ab)c = a+b+c+ab+ac+bc+abc\).
  • Next, compute \(a \cdot (b \cdot c)\): Begin with \(b \cdot c = b + c + bc\). Next, calculate \(a \cdot (b + c + bc) = a + (b+c+bc) + a(b+c+bc) = a+b+c+ab+ac+abc\).
Since both expressions are equal, \(*\) satisfies the associative property within the set \(S\).
Identity Element
An identity element in a set with a binary operation is an element that, when combined with any element of the set using the operation, leaves the element unchanged. In our set \(S\), we need an element \(e\) such that \(a \cdot e = a\) and \(e \cdot a = a\) for all \(a\) in \(S\).
  • Consider \(a \cdot e = a + e + ae = a\). This simplifies to \(e(1 + a) = 0\). Given that \(a eq -1\), \(e = 0\).
  • Therefore, 0 acts as the identity element in \((S, *)\) because \(a \cdot 0 = a\) and \(0 \cdot a = a\) for any element \(a\) in \(S\).
Inverse Element
For a group, every element must have an inverse such that combining the element and its inverse result in the identity element. In \((S, *)\), the identity element is 0, thus for any element \(a\) in \(S\), its inverse \(b\) must satisfy \(a \cdot b = 0\).
  • Starting from \(a + b + ab = 0\), solving for \(b\) yields \(b = \frac{-a}{1+a}\).
  • Since \(a eq -1\), \(1+a eq 0\), ensuring \(b\) is defined and within \(S\).
Thus, the inverse element of \(a\) in \((S, *)\) is \(b = \frac{-a}{1+a}\), achieving the condition \(a \cdot b = 0\).

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Most popular questions from this chapter

In Exercises 11 through 18, determine whether the given set of matrices under the specified operation, matrix addition or multiplication, is a group. Recall that a diagonal matrix is a square matrix whose only nonzero entries lie on the main diagonal, from the upper left to the lower right corner. An upper- triangular matrix is a square, matrix with only zero entries below the main diagonal. Associated with each \(n \times n\) matrix \(A\) is a number called the determinant of \(A\), denoted by \(\operatorname{det}(A)\). If \(A\) and \(B\) are both \(n \times n\) matrices, then \(\operatorname{det}(A B)=\operatorname{det}(A) \operatorname{det}(B)\), Also, \(\operatorname{det}\left(I_{n}\right)=1\) and \(A\) is invertible if and only if \(\operatorname{det}(A) \neq 0\). All \(n \times n\) diagonal matrices under matrix addition.

In Exercises 11 through 18, determine whether the given set of matrices under the specified operation, matrix addition or multiplication, is a group. Recall that a diagonal matrix is a square matrix whose only nonzero entries lie on the main diagonal, from the upper left to the lower right corner. An upper- triangular matrix is a square, matrix with only zero entries below the main diagonal. Associated with each \(n \times n\) matrix \(A\) is a number called the determinant of \(A\), denoted by \(\operatorname{det}(A)\). If \(A\) and \(B\) are both \(n \times n\) matrices, then \(\operatorname{det}(A B)=\operatorname{det}(A) \operatorname{det}(B)\), Also, \(\operatorname{det}\left(I_{n}\right)=1\) and \(A\) is invertible if and only if \(\operatorname{det}(A) \neq 0\). All \(n \times n\) diagonal matrices with all diagonal entries 1 or \(-1\) under matrix multiplication.

All \(n \times n\) upper-triangular matrices with determinant 1 under matrix multiplication.

We can also consider multiplication : \(n\) modulo \(n\) in \(Z_{n}\). For example, \(5+6=2\) in \(\mathbb{Z}\) - because \(5 \cdot 6=30=\) \(4(7)+2\). The set \(\\{1,3,5,7\\}\) with multiplication \(+\) modulo 8 is a group. Give the table for this group,

Mark each of the following true or false. _____a. A group may have more than one Identity element. _____b. Any two groups of three elements are isomorphic. _____c. In a group, each linear equation has a solution. _____d. The proper attitude toward a detinition is to memorize it so that you can reproduce it word for word as in the text. _____e. Any definition a person gives for a group is correct provided that everything that is a group by that person's definition is also a group by the definition in the text. _____f. Any definition a person gives for a group is correct provided he or she can show that cverything that satisfies the definition satisfies the one in the text and conversely. _____g. Every finite group of at most three elements is abelian. _____h. An equation of the form \(a+x * b=c\) always has a unique solution in a group. _____i. The empty set can be considered a group. _____j. Every group is a binary algebraic structure.
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