Mathematical Systems

• This work was started by Euler and Gauss.
• But Galois, in his short life, made the first major contributions.
• A mathematical system is a set of elements along with a binary operation on those elements.
  • The integers along with addition is an example
  • Addition take two elements of the set of integers and returns a element of the set of integers.
  • It is a binary operation because it takes two element.
    • Multiplication is also a binary operation.
    • The intersection of two sets is a binary operation.
    • If and only if is a binary operation.
    • set complement is not a binary operation.
    • not is not a binary operation.
  • Subtraction on the natural numbers is a mathematical system, but.
  • Division on the integers is a mathematical system
  • but they are strange in that it does not return an element of the original set.
• Once we have a system, we start to investigate properties the system possesses
• Commutative property
  • The commutative Property of Addition : a+b = b+a
  • The commutative property of Multiplication : a*b = b*a
  • Is subtraction commutative? a-b = b- a
  • Is AND commutative (p ∧ q ) = (q ∧ p)
  • Is If-Then commutative (p → q ) = (q → p)?
• Associative property
  • Addition: (a+b)+c = a+(b+c)
  • Multiplication (ab)c = a(bc)
  • Is subtraction associative? (a-b)-c = a-(b-c) (try 1 2 3)
  • Is AND associative
• Closure:
  • If a binary operation is performed on any tow elements of a set and the result is an element of the set, then the set is closed (or has closure) under the binary operation.
  • All of the boolean operators exhibit closure.
  • The natural numbers are closed under addition.
  • The natural numbers are not closed under subtraction. Why? (4-5)
  • 4-5 is a counterexample to the claim that the natural numbers are closed under subtraction.
• Identity Element
  • An identity element is an element in a set such that when a binary operation is performed on it, and any given element of the set, the result is the given element.
  • 0 is the identity element on the integers under addition 0+x =x
  • 1 is the identity element on the natural numbers under multiplication 1*x = x
  • T is the identity element on the booleans under and t ∧ x = x (t∧ t =t, t∧f = f)
  • What is the identity element on the booleans under or?
  • Is there an identity element on the booleans under if-then?
• Inverse
  • When a binary operation is performed on two elements in a set and the result is the identity element for the binary operation, each element is said to be the inverse of the other.
  • For the integers under addition, what gives us 0 (the identity element)
  • 4 + (-4) = 0, -7 + (--7) = 0,
  • So if a is an integer, -a is the additive inverse.
  • Is there an inverse for the integers under multiplication?
  • How about the booleans under and
• A group
  • A set of elements along with an operation is a group if
    1. The set is closed under the operation
    2. The identity element exists on the set for the operation
    3. The inverse of every element exists on the set for the operation
    4. The Set of elements is associative under the operation.
  • The integers under addition is an example of a group.
  • The real numbers under addition is an example of a group.
  • Are the rational numbers under multiplication a group? no, 0 has no multiplicative inverse.
• A commutative group is a group which is commutative under the given operation.
• Exercises
  • Is the set of integers a commutative group under addition?
  • Is the set of positive integers a commutative group under addition.
  • Is the set of positive integers a group under (multiplication, subtraction)
  • Are the booleans a group under AND
  • Are sets a a group under intersection?