### Algebra and Analysis for Engineers and Scientists by Anthony N. Michel, Charles J. Herget

• March 23, 2017
• System Theory
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By Anthony N. Michel, Charles J. Herget

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Extra resources for Algebra and Analysis for Engineers and Scientists

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Are monoids. " • The most important special type of semigroup that we will encounter in this chapter is the group. 18. Definition. , a group is a semigroup, ;X { invertible. The set R of real numbers with the operation of addition is an example of a group. The set of real numbers with the operation of multiplication does not form a group, since the number zero does not have an inverse relative to multiplication. However, the latter system is a monoid. } is a group. Groups possess several important properties.

Y The subset of X, {x :X (x, y) E E p, Y E )Y , is called the domaiD or p. The subset of Y {y :Y (x, y) E E p, X EX ) , is called the ruge of p. X The relation p- I is called the inverse relation of p. Note that whereas the inverse of a function does not always exist, the inverse of a relation does always exist. Next, we consider equivalence relations. either (x, y) E P or (x, y) i p, but not both. J/y. 5. DefiDition. X (i) If x P x for all x E ,X then p is said to be reflexive; Chapter 1 26 I uF ndtzmental Concepts (ii) if x P y implies y p x for all x, Y E p, then p is said to be symmetric; and (iii) if for all x, y, Z E ,X X PY and y p Z implies x p ,z then p is said to be traositive.

J/y. 5. DefiDition. X (i) If x P x for all x E ,X then p is said to be reflexive; Chapter 1 26 I uF ndtzmental Concepts (ii) if x P y implies y p x for all x, Y E p, then p is said to be symmetric; and (iii) if for all x, y, Z E ,X X PY and y p Z implies x p ,z then p is said to be traositive. 6. Example. Let R denote the set of real numbers. The relation in R given by {(x, y): x < y} is transitive but not reflexive and not symmetric. y} is symmetric but not reflexive and The relation in R given by {(x, y): x not transitive.