or each of the following functions, state whether it is injective, surjective, and/or bijective, and why.(a) The function f (n) = 2n, mapping from integers to integers.(b) The function q(ϕ), with codomain N≥0, which maps any formula of predicate logic to the number of quantifiersin that formula.
Question
or each of the following functions, state whether it is injective, surjective, and/or bijective, and why.(a) The function f (n) = 2n, mapping from integers to integers.(b) The function q(ϕ), with codomain N≥0, which maps any formula of predicate logic to the number of quantifiersin that formula.
Solution
(a) The function f(n) = 2n, mapping from integers to integers, is injective but not surjective.
Injective: A function is injective (or one-to-one) if every element of the domain maps to a unique element in the codomain. In this case, for every integer n, there is a unique integer 2n. No two integers map to the same integer under this function, so it is injective.
Surjective: A function is surjective (or onto) if every element of the codomain is the image of at least one element of the domain. In this case, the codomain is the set of all integers. However, only even integers are the image of the function f(n) = 2n. Odd integers are not the image of any integer under this function, so it is not surjective.
Bijective: A function is bijective if it is both injective and surjective. Since this function is not surjective, it is not bijective.
(b) The function q(ϕ), with codomain N≥0, which maps any formula of predicate logic to the number of quantifiers in that formula, is neither injective nor surjective without additional information.
Injective: Without additional information, we cannot determine if this function is injective. It is possible for different formulas of predicate logic to have the same number of quantifiers, in which case the function would not be injective.
Surjective: Without additional information, we cannot determine if this function is surjective. If there exists a number in N≥0 that is not the number of quantifiers in any formula of predicate logic, then the function is not surjective.
Bijective: Since we cannot determine if this function is injective or surjective, we cannot determine if it is bijective.
Similar Questions
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