If A and B are sets, the cartesian product of A and B, denoted A x B ("A times B"),is the set consisting of all ordered pairs (a,b) with a € A and b € B; the elements (a₁, b₁)and (a2, b₂) of A x B are equal if and only if a₁ = a2 and b₁ b₂. (One can define cartesianproducts of more than two sets similarly.) Note that, if either A or B is the empty set, i.e.,if either A = 0 or B = 0, then A x B = 0 (the symbol always denotes the empty set, andshould not be used for the number 0).A function (or map, or mapping) from A to B is a subset f of A x B with the propertythat, for every a € A, there is exactly one b E B such that (a, b) € f; it is customary todenote this b by f(a), and to refer to f(a) as the image of a under f, or the value of f at a.To indicate that f is a function from A to B, we write f : A → B; the set A is the domainof f, and B is the codomain of f; the range of f is the set f(A) defined byf(A) = {f(a): a € A}.Two functions are equal if and only if they have the same domain, the same codomain, andthe same value at each element of their common domain. This means that, for instance, ifwe define functions f: R→ R and g: R→ [0, ∞) by f(x) = x² = g(x) for all x € R, thenfg because f and g have distinct codomains; note however that f(R) = g(R) = [0, ∞0),so f and g have the same range. Despite the definition, one almost never thinks of a functionas a set of ordered pairs; instead one usually interprets a function f : A → B as a "rule"or "procedure" for assigning to each element a in its domain a unique element f(a) in itscodomain.Some remarks are in order. If f: A → B is a function, it is not correct to say that f(a)is a function; indeed, f(a) is an element of the codomain B, while f itself is (by definition)a subset of Ax B. So, for example, we do not say anything like "consider the function z²,"and we would instead say "consider the function f: R → R defined by f(x) = x²." Thisexample highlights a second issue which, like the first, is often ignored in first-year calculus:the domain and codomain are essential components of a function, and should always beexplicitly specified as part of the function's definition. This is why, above, we did not say"consider the function f defined by f(x) = x²;" we explicitly indicated the intended domainand codomain when we defined the function.1. Suppose we would like to define a function f : Q→ Z by the formulaf(r) = a +b,where a, b € Z are integers with b 0 such that r = a/b (every rational number can berepresented as such a fraction, by definition). This is an instance of a formula which does notyield a well-defined function. What do you think is meant by this phrase? What propertyin the definition of a function from Q to Z does the displayed formula fail to satisfy? Justifyyour answer with an explicit numerical example.

Question

If A and B are sets, the cartesian product of A and B, denoted A x B (&#34;A times B&#34;),is the set consisting of all ordered pairs (a,b) with a € A and b € B; the elements (a₁, b₁)and (a2, b₂) of A x B are equal if and only if a₁ = a2 and b₁ b₂. (One can define cartesianproducts of more than two sets similarly.) Note that, if either A or B is the empty set, i.e.,if either A = 0 or B = 0, then A x B = 0 (the symbol always denotes the empty set, andshould not be used for the number 0).A function (or map, or mapping) from A to B is a subset f of A x B with the propertythat, for every a € A, there is exactly one b E B such that (a, b) € f; it is customary todenote this b by f(a), and to refer to f(a) as the image of a under f, or the value of f at a.To indicate that f is a function from A to B, we write f : A → B; the set A is the domainof f, and B is the codomain of f; the range of f is the set f(A) defined byf(A) = {f(a): a € A}.Two functions are equal if and only if they have the same domain, the same codomain, andthe same value at each element of their common domain. This means that, for instance, ifwe define functions f: R→ R and g: R→ [0, ∞) by f(x) = x² = g(x) for all x € R, thenfg because f and g have distinct codomains; note however that f(R) = g(R) = [0, ∞0),so f and g have the same range. Despite the definition, one almost never thinks of a functionas a set of ordered pairs; instead one usually interprets a function f : A → B as a &#34;rule&#34;or &#34;procedure&#34; for assigning to each element a in its domain a unique element f(a) in itscodomain.Some remarks are in order. If f: A → B is a function, it is not correct to say that f(a)is a function; indeed, f(a) is an element of the codomain B, while f itself is (by definition)a subset of Ax B. So, for example, we do not say anything like &#34;consider the function z²,&#34;and we would instead say &#34;consider the function f: R → R defined by f(x) = x².&#34; Thisexample highlights a second issue which, like the first, is often ignored in first-year calculus:the domain and codomain are essential components of a function, and should always beexplicitly specified as part of the function's definition. This is why, above, we did not say&#34;consider the function f defined by f(x) = x²;&#34; we explicitly indicated the intended domainand codomain when we defined the function.1. Suppose we would like to define a function f : Q→ Z by the formulaf(r) = a +b,where a, b € Z are integers with b 0 such that r = a/b (every rational number can berepresented as such a fraction, by definition). This is an instance of a formula which does notyield a well-defined function. What do you think is meant by this phrase? What propertyin the definition of a function from Q to Z does the displayed formula fail to satisfy? Justifyyour answer with an explicit numerical example.

Accepted Answer

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