MAT 102 Mathematical Systems & Groups

Definition A mathematical system (�, ∗) consists of a set � with elements �, �, �, �, … together with a binary operation ∗ that combines any two elements in � to create a new element (e.g. � ∗ �, � ∗ �, � ∗ �, etc.). Note that the set � can be either finite or infinite.

Examples

• (ℝ, ×) is the system of real numbers (ℝ) under multiplication. • (ℕ, −) is the system of counting numbers, or naturals, (ℕ) under subtraction. • The set of naturals 1 through 12 under “clock-12” addition or subtraction. These systems are generalized for any finite set (clocks with � hours) using mod � modular arithmetic. • The set {�, �} of odd and even integers under the usual operations of arithmetic. • The set {�, �} of truth values for statements under the “and” (conjunction) or “or (disjunction) connectives of propositional logic. • The set of symmetries of a square on a plane (i.e. the geometrical movements that keep the square unchanged) under rotations or reflections about an axis.

Number Sets

Below is a summary of some number sets commonly featured in mathematical systems.

• ℙ = {2, 3, 5, 7, 11, 13, 17, 19, … } is the set of prime numbers. • ℕ = {1, 2, 3, 4, 5, … } is the set of naturals, or counting numbers. • � = {0, 1, 2, 3, 4, 5, … } is the set of whole numbers. • ℤ = {1, 2, 3, 4, 5, … } is the set of integers. • ℚ is the set of rational numbers, or fractions. These are the numbers whose decimal digits either terminate (e.g. 2⁄5 = 0.4) or have an unending but repeating pattern (e.g. 2⁄9 = 0.222 …). • � is the set of irrational numbers. These are the numbers whose decimal digits never terminate and do not have a repeating pattern (e.g. √2, �, 0.1011011101111…). • ℝ is the set of real numbers. These are all the rationals together with all the irrationals. They are the numbers that can be placed on a number line extending without bounds from left (−∞) to right (+∞).

Properties

Consider adding two numbers � and � from any of the number sets listed above. Many important properties turn out to be true for such a system. For example, the operation of addition is commutative since � + � = � + � and the number 0 acts as the additive identity since � + 0 = 0 + � = �. Other properties can also be verified for this system, such as associativity or the existence of inverses for � or � (provided the set of numbers is at least as big as the integers).

Below are presented five important properties of mathematical systems. For all we consider the set of elements �, �, �, �, … of a set � under some binary operation ∗ that is well-defined.

1. Closure � is closed under the operation ∗ if, for any two elements �, � in �, the element � ∗ � is also in �. For example, the naturals are closed under addition since adding any two naturals always results in another, bigger, natural (e.g. 2 + 5 = 7). This same set, however, is not closed under subtraction since this operation can lead to negative integers (e.g. 2 − 5 = −3). Any instance, such as this last one, resulting in an element outside of � is called a counter-example.

2. Identity � has an identity element � if, for any element � in �, � ∗ � = � ∗ � = �. In other words, an identity is an element of � that, when combined through the operation ∗ with any other element in �, preserves that element. For example, 12 is the identity of the set of hours under “clock-12” (mod 12) addition since ℎ + 12 ≡ ℎ ��� 12 for ℎ = 1, 2, 3, … . , 12. In the usual systems of arithmetic, 0 is the additive identity and 1 is the multiplicative identity.

3. Inverses If an element � in � is such that � ∗ � = � ∗ � = �, then this element is called the inverse of � and (�, �) is called an inverse pair. For example, (7, 5) is an inverse pair of hours in the “clock-12” additive system since 5 + 7 ≡ 7 + 5 ≡ 12 ��� 12. With respect to the usual operations of arithmetic, the additive inverse of a number � is −� (the opposite of �) since � + (−�) = � − � = 0 and the multiplicative inverse of a non- zero number � is (the reciprocal of �) since � × = = 1. Note that the number 0 fails to have a multiplicative inverse since division by zero is undefined.

4. Associativity If for any three elements �, �, � in �, (� ∗ �) ∗ � = � ∗ (� ∗ �), the operation ∗ is said to be associative. This property is clearly true for addition and multiplication. For example, (1 + 2) + 3 = 3 + 3 = 6 and 1 + (2 + 3) = 1 + 5 = 6. However, the property fails for division or subtraction. For example, [(−2) − (−4)] − 5 = 2 − 5 = −3 yet (−2) − [(−4) − 5] = (−2) − (−9) = 7.

5. Commutativity If for any two elements �, � in �, � ∗ � = � ∗ �, the operation ∗ is said to be commutative. This property is also clearly true for addition and multiplication. For example, × 10 = 10 × = 4. Note again that this property fails for division or subtraction. For example, 3 ÷ 6 = 0.5 is different from 6 ÷ 3 = 2.

Definition A mathematical system (�, ∗) that satisfies properties 1 through 4 is called a group. A group that also satisfies property 5 is called an Abelian (or commutative) group.

What this definition implies then is that a group (�, ∗) must have a set � that is closed under ∗ and includes an identity element. Moreover, every element in � must have an inverse and the operation ∗ must be associative.

Exercise 1

Is the set {1, 2, 3} under ��� 4 multiplication a group?

Exercise 2

Is the set of truth values {�, �}, where � is “True” and � is “False”, for statements under the logical operation of conjunction a group?

In logic, if � and � are basic statements that are either true (�) or false (�), then the conjunction of � and � is the compound statement “� and �.” According to the truth table for conjunction (see below), “� and �” is only true when both statements � and � are true.

� � � and � T T T T F F F T F F F F

Exercise 3

Are the following arithmetical systems groups? If yes, show that all four group properties hold for the system and check whether the group is Abelian (commutative). If no, show which of the four properties fail for the system.

A) The set of whole numbers under subtraction. B) The set of positive rational numbers (i.e. fractions) under multiplication. C) The set of even integers under addition. D) The set of real numbers under division.

Solution – Exercise 1

This system is not a group since it is not closed. As shown in the table below, 2 × 2 ≡ 0 ��� 4 and 0 is not in the set of numbers {1, 2, 3}.

��� � 1 2 3 multiplication 1 1 2 3 2 2 0 2 3 3 2 1

Checking all other properties, we find the following:

• The set does have the identity 1 since any number 1, 2, or 3 multiplied by 1 mod 4 is itself (check this in the table with row 1 and column 1).

• The numbers 1 and 3 have themselves as inverses, but 2 fails to have an inverse since the congruence 2 × � ≡ 1 ��� 4 has no solution � in the set {1, 2, 3}. This can be seen in the table by noting that 1 does not appear in the second row or the second column.

• Associativity holds for this operation, though it is difficult to prove it. Here is an example that illustrates this property:

(2 × 3) × 1 ≡ 2 × 1 ≡ 2 ��� 4 and 2 × (3 × 1) ≡ 2 × 3 ≡ 2 ��� 4.

• Commutativity holds for this operation (e.g. 2 × 3 ≡ 3 × 2 ≡ 2 ��� 4) and it can be checked by noting that the numbers in the 3 × 3 table are symmetric with respect to the so- called main diagonal of entries (1, 0, 1) running from top-left to bottom-right.

Solution – Exercise 2

The set of truth values under the logical operation of conjunction is not a group. As checked below, one of the elements in the set fails to have an inverse while all other group properties hold.

§ Closure The set is closed since the truth table for conjunction only contains the truth values T and F.

§ Identity The system has the identity T since T and T = T (row 1) and F and T = F (row 3).

§ Inverses Checking for inverses, we see that the inverse of T is T since T and T = T (row 1). However, F fails to have an inverse since a conjunction with a false component is never true!

§ Associativity It is easy to check that the logical operation of conjunction is associative.

Solution – Exercise 3

A) The set of whole numbers under subtraction is not a group. As checked below, it violates all four group properties.

§ Closure The system is not closed. For example: 1 − 3 = −2, but −2 is not a whole number.

§ Identity & Inverses The system has no identity and, therefore, none of its elements have inverses. To see that there is no identity here, argue by contradiction. Assume � is the identity. Then, for any whole number �, we’d have � − � = � − � = �. But that would imply that � = 0 = �… Clearly this is not a possibility whenever � is positive. Therefore, there is no identity!

§ Associativity The operation of subtraction is not associative. In general, for any three whole numbers �, � and �, we have (� − �) − � = � − � − � ≠ � − (� − �) = � − � + �. For example, (5 − 3) − 1 = 2 − 1 = 1 ≠ 5 − (3 − 1) = 5 − 2 = 3.

B) The set of positive fractions under multiplication is an Abelian group. As checked below, it satisfies all five properties for commutative groups.

§ Closure The system is closed since the product of any two positive fractions is also a positive fraction.

§ Identity The system has the usual multiplicative identity 1 since × 1 = 1 × = for every positive fraction .

§ Inverses Every positive fraction has its reciprocal as its inverse since × = 1.

§ Associativity & Commutativity The operation of multiplication is associative since for any three positive ×× fractions , , and , we have × × = × × = . For example, ×× × × = × × = . It is also commutative. (Check this.)

C) The set of even integers ℤ = {0, ±2, ±4, ±6, … } under addition is an Abelian group. As checked below, it satisfies all five condition for commutative groups.

§ Closure The system is closed since adding two even integers always yields another even integer. For example: −6 + 2 = −4.

§ Identity The system has the usual additive identity 0 since 2� + 0 = 0 + 2� = 2� for any even integer 2� (where � is another integer).

§ Inverses Every even integer 2� has its opposite −2� as its inverse since 2� + (−2�) = 0. For example, −6 + 6 = 0.

§ Associativity & Commutativity Since the operation of addition is associative and commutative for integers, it must also have the same properties for even integers. For example, −2 + (−4 + 10) = −2 + (−4) + 10 = 4 and −2 + 4 = 4 − 2 = 2.

D) The set of real numbers (i.e. all the numbers that can be placed on a number line) under division is not a group. As checked below, it fails 3 out of the 4 group conditions.

§ Closure The system is closed since the quotient of any two real numbers is also a real number. Note that the operation of division does not include cases where the denominator is zero since dividing any number by zero is not defined as a valid operation in arithmetic.

§ Identity & Inverses The system has no identity and so none of its elements have inverses. Arguing by contradiction, if the system had an identity �, then it would satisfy the following equations for any non-zero real number �: = = �. But this would imply that � = 1 = �. This is clearly not true for any real number � ≠ 1. Thus, there is no identity!

§ Associativity The operation of division is not associative. Here’s a counterexample:

(1 ÷ 2) ÷ 10 = = 0.05 ≠ 1 ÷ (2 ÷ 10) = 5.