X


Set Operations







Set Identities





Functions




Domain and Codomain of a function





Q. What are the domain, codomain, and range of the function that assigns grades to students described in the first paragraph of the introduction of this section?



Q. Let R be the relation with ordered pairs (Abdul, 22), (Brenda, 24), (Carla, 21), (Desire, 22), (Eddie, 24), and (Felicia, 22). Here each pair consists of a graduate student and this student’s age. Specify a function determined by this relation



Q. Let f be the function that assigns the last two bits of a bit string of length 2 or greater to that string. For example, f(11010) = 10. Then, the domain of f is the set of all bit strings of length 2 or greater, and both the codomain and range are the set {00, 01, 10, 11}.





Q. The domain and codomain of functions are often specified in programming languages. For instance, the Java statement int floor(float real){...} and the C++ function statement int function (float x){...}



Definition 3: Function



Q. f1 and f2 be functions from R to R such that f1(x) = x2 and f2(x) = x - x2. What are the functions (f1 + f2)(x) and (f1 . f2)(x)


Step 2:


DEFINITION 4




One-to-One and Onto Functions






Q. Determine whether the function f from {a, b, c, d} to {1, 2, 3, 4, 5} with f(a) = 4, f(b) = 5, f(c) = 1, and f(d) = 3 is one-to-one



Q. Determine whether the function f(x) = x2 from the set of integers to the set of integers is one-to-one.



Increasing and Decreasing functions :



Surjective Functions





Q. Let f be the function from {a, b, c, d} to {1, 2, 3} defined by f(a) = 3, f(b) = 2, f(c) = 1, and f(d) = 3. Is f an onto function?



Q. Is the function f(x) = x2 from the set of integers to the set of integers onto?



Q. Is the function f(x) = x + 1 from the set of integers to the set of integers onto?





Function : one-to-one correspondence, or a bijection



Q. Let f be the function from {a, b, c, d} to {1, 2, 3, 4} with f(a) = 4, f(b) = 2, f(c) = 1, and f(d) = 3. Is 'f' a bijection?



The identity function on A





Suppose that f : A → B.


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