Section 13.1

1. Math in Our World Section 13.1 Mathematical Systems and Groups

2. Learning Objectives • Use an operation table to perform the operation in a mathematical system. • Determine which properties of mathematical systems are satisfied by a given system. • Decide if a mathematical system is a group.

3. Mathematical Systems A mathematical system consists of a finite or infinite set of symbols and at least one operation. When you’re driving on city streets and come to a four-way intersection, you have four choices: right, left, straight, or U-turn. In this case, the four symbols are R, L, S, and U, representing your four choices at the intersection. Consider what happens if you make two of those choices consecutively.

4. Mathematical Systems Consider what happens if you make two of those choices consecutively. If you turn right and then left, you end up going in the same direction you started in, so this corresponds to going straight. If you make a U-turn, then turn left, you’ll end up in the same direction as if you had turned right.

5. Mathematical Systems The symbols represent the combinations we already mentioned: R + L = S, and U + L = R. The table represents a finite mathematical system because there are a finite number of symbols with an operation and is called an operation table. A system with infinitely many symbols and an operation is called an infinite mathematical system.

6. EXAMPLE 1 Using an Operation Table Use the table to find the result of each operation and describe what it means physically. (a) L + S (b) R + R (c) U + S (d) (S + U) + R

7. EXAMPLE 1 Using an Operation Table SOLUTION (a) The symbol in row L and column S is L, so L + S = L. If you turn left, then go straight, the direction is the same as just turning left.

8. EXAMPLE 1 Using an Operation Table SOLUTION (b) The symbol in row R and column R is U, so R + R = U. Turn right twice and your direction is the same as if you had made a U-turn.

9. EXAMPLE 1 Using an Operation Table SOLUTION (c) U + S = U; a U-turn followed by straight is the same direction as the U-turn itself.

10. EXAMPLE 1 Using an Operation Table SOLUTION (d) (S + U) + R First, we find that S + U = U; then U + R = L. If you follow going straight and making a U-turn with a right turn, it’s the same direction as just turning left.

11. Five Properties of Mathematical Systems • Closure Property • For a system to be closed under an operation, when the operation is performed on any symbol, the result must be another symbol in the system. The system of “turns” satisfies this property, and is called a closed system because any combination of two turns results in one of the four basic turns.

12. Five Properties of Mathematical Systems 2. Commutative Property A system is commutative if the order in which you perform the operation doesn’t matter. More formally, a system is commutative if for any two symbols a and b in a system with some operation *, a * b = b * a. The system of “turns” is commutative because the result of any two consecutive turns is the same regardless of the order you perform them in.

13. Five Properties of Mathematical Systems 2. Commutative Property Notice that the table is symmetric about the diagonal from upper left to lower right: that is, the bottom left and upper right triangular portions are mirror images.

14. Five Properties of Mathematical Systems 3. Associative Property A system has the associative property if for any a, b, c in a system with operation *, (a * b) * c = a * (b * c). There really isn’t a quick way to show that a system is associative: you would have to try every possible combination of three symbols. Typically, we’ll check a few examples to try and get an idea of whether or not a system is associative.

15. Five Properties of Mathematical Systems 4. Identity Property A system has the identity property if there is a symbol a in the system so that a * b = b * a = b for any other symbol b. In short, if there’s a symbol that leaves all the others unchanged when combined using the operation, that symbol is called the identity element for the system. For our system of “turns,” you can see that S is the identity element.

16. Five Properties of Mathematical Systems 5. Inverse Property When a system has an identity element, the next question to consider is whether every symbol has another symbol that produces the identity element under the operation. That is, for any a in the system, is there always an inverse element b so that a * b is the identity element? If there is, the system satisfies the inverse property.

17. EXAMPLE 2 Identifying the Properties of a Finite Mathematical System Which properties does the system defined by the given table exhibit?

18. EXAMPLE 2 Identifying the Properties of a Finite Mathematical System SOLUTION Closure property: Since every element in the body of the table is in the set the system is defined on, namely, {0, 1, 2, 3}, the system is closed. Commutative property: Since the system is not symmetric with respect to the diagonal, it does not have the commutative property. For example, 0 & 1 = 3, but 1 & 0 = 1.

19. EXAMPLE 2 Identifying the Properties of a Finite Mathematical System SOLUTION Associative property: Let’s try some examples. (1 & 2) & 3 = 1 & 3 = 0 1 & (2 & 3) = 1 & 3 = 0 This one works! Let’s try: (1 & 0) & 3 = 1 & 3 = 0 1 & (0 & 3) = 1 & 1 = 2 We found a counterexample, so the system is not associative.

20. EXAMPLE 2 Identifying the Properties of a Finite Mathematical System SOLUTION Identity property: There is an identity element, 2, since 2 & x = x & 2 = x for all x = {0, 1, 2, 3} Inverse Property: 0 and 0 are inverses, since 0 & 0 equals the identity. 1 and 1 are inverses, 2 and 2 are inverses, and 3 and 3 are inverses. Since every element has an inverse, the system has the inverse property. Therefore, the system exhibits closure, identity, and inverse properties.

21. Groups and Abelian Groups A mathematical system is said to be a group if it has closure, associative, identity, and inverse properties. A mathematical system is said to be an Abelian group if, in addition to closure, associative, identity, and inverse properties, it also has the commutative property.

22. EXAMPLE 3 Determining If a Mathematical System is a Group Do the natural numbers under the operation of addition form a group? An Abelian group? SOLUTION The natural numbers are given by the set N = {1, 2, 3, 4,… }. Closure: The natural numbers are closed under addition; that is, the sum of any two natural numbers is a natural number. Therefore the system is closed. Associative: The associative property of addition holds for all real numbers, and since natural numbers are real numbers, it holds for the natural numbers as well. So the system is associative.

23. EXAMPLE 3 Determining If a Mathematical System is a Group SOLUTION Identity: There is no identity element in the set of natural numbers (the identity element would be 0 under addition, but this element is not in N). Inverse: Since there is no identity element, there can be no inverse. Therefore the inverse property does not hold. Since the system does not have the four properties required to be a group, it is not a group. Since it is not a group, it is also not an Abelian group.

24. EXAMPLE 4 Determining If a Mathematical System is a Group Does the set { - 1, 1} form a group under the operation of multiplication? An Abelian group? SOLUTION Closure: The product of any two elements in the set is also an element of the set, therefore the system is closed. Associative: The associative property holds for all elements in the set. Identity: The identity element is 1. Inverse: -1 and - 1 are inverses as are 1 and 1. Therefore each element has an inverse, so the inverse property holds. Since the system has all four properties, it is a group. And, since the commutative property holds, it’s also an Abelian group.