Decide whether the given set is a group with respect to the indicated operation. If it is not a group, state a condition in the definition of a group that fails to hold. The set of all positive irrational numbers with operation multiplication.
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A group has to satisfy the following operations:
Closure - the operation has to take elements of the set to the set. With irrational numbers, this fails. For example, `sqrt2 times sqrt2=2` has two irrational numbers that do not satisfy the closure requirement.
Associativity - if a, b, c are elements of the set, then (ab)c=a(bc). This is satisfied with the set.
Identity element - there is an identity element from the set with the operation. This is not satisfied with the set of positive irrational numbers, since 1 is rational.
Inverse element - this is satisifed with irrational numbers and multiplication.
The set is not a group and fails to hold with closure and the identity element.
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