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Handout Friday, 8/29
We will continue our discussion on operations and their properties.
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First, an operation is a function from pairs of elements to elements of that set. What does it mean to be a function? List the properties that make it a function in this case.
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Is subtraction an operation? Is division? Well, on what set? To be an operation, you are always an operation on some set, and that set must be closed under that operation. In other words, if you apply the operation to elements of the set, you must get an element of the set.
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Consider the operation \(*\) on \(\Z\) (the set of integers) given by \(a * b = a + 2b + ab\text{.}\) Check this really is an operation.
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Last time we saw that in order to solve equations, we needed to have some basic properties. Let \(*\) be an operation on a set \(S\text{.}\)
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We say \(*\) is associative provided that for all \(a\text{,}\) \(b\text{,}\) and \(c\) in the set, \(a * (b * c) = (a * b) * c\text{.}\)
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We say \(*\) is commutative provided that for all \(a\) and \(b\) in the set, \(a * b = b * a\text{.}\)
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We call an element \(e\) in \(S\) an identity for \(*\) if for all \(a\) in \(S\text{,}\) \(e * a = a * e = a\text{.}\)
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We say that an element \(a \in S\) has an inverse for \(*\) if there exists an element \(b \in S\) such that \(a * b = b * a = e\text{,}\) where \(e\) is the identity element for \(*\text{.}\)
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Sets for which there is an operation that is associate, contains an identity, and for which every element has an inverse, are called groups.
Let’s look at some examples of supposed operations to investigate their properties.
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Is division an operation on \(\Z\text{?}\) No, because the set is not closed under division.
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Is subtraction an operation on \(\Z\text{?}\) Yes. But note that it is not associative or commutative.
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What about the operation \(*\) on \(\Z\) given by \(a * b = a + 2b - ab\text{?}\) It is an operation. Is it associative? Commutative? Does it have an identity? Do all elements have inverses?
