Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter B.3, Problem 1E
a.
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b.
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4. Let A, B and C be sets, and let g: A → B and f: B → C be functions. Prove:
(i) If f, g are surjective, then so is fog.
(ii) If f, g are surjective and at the same time fog is not, then fog is injective.
Suppose f : A → B is an injective function. Show that there is a function g : B → A such thatg ◦ f = idA. Here, idA : A → A is the identity function on A, i.e., the function that satisfies idA(a) = afor all a ∈ A. (Assume f is non-trivial, i.e., A ̸= ∅!)
Let A = {a, b, c, d, e} and B = {1, 2, 3, 4, 5, 6, 7, 8}. How many functions f : A → B(a) ... are injective?(b) ... are not injective?(c)... are such that f(a) = f(b) = f(c)?(d) ... are such that exactly three elements of A have 8 as an image?(e) ... are surjective?
Chapter B Solutions
Introduction to Algorithms
Ch. B.1 - Prob. 1ECh. B.1 - Prob. 2ECh. B.1 - Prob. 3ECh. B.1 - Prob. 4ECh. B.1 - Prob. 5ECh. B.1 - Prob. 6ECh. B.2 - Prob. 1ECh. B.2 - Prob. 2ECh. B.2 - Prob. 3ECh. B.2 - Prob. 4E
Ch. B.2 - Prob. 5ECh. B.3 - Prob. 1ECh. B.3 - Prob. 2ECh. B.3 - Prob. 3ECh. B.3 - Prob. 4ECh. B.4 - Prob. 1ECh. B.4 - Prob. 2ECh. B.4 - Prob. 3ECh. B.4 - Prob. 4ECh. B.4 - Prob. 5ECh. B.4 - Prob. 6ECh. B.5 - Prob. 1ECh. B.5 - Prob. 2ECh. B.5 - Prob. 3ECh. B.5 - Prob. 4ECh. B.5 - Prob. 5ECh. B.5 - Prob. 6ECh. B.5 - Prob. 7ECh. B - Prob. 1PCh. B - Prob. 2PCh. B - Prob. 3P
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- Q. Let A = {a, b, c, d, e} and B = {1, 2, 3, 4, 5, 6, 7, 8}. How many functions f : A → B(a) ... are injective?(b) ... are not injective?(c) ... are such that f(a) = f(b) = f(c)?(d) ... are such that exactly three elements of A have 8 as an image?(e) ... are surjective?arrow_forwardSuppose that fis a function from A to B, where A and B are finite sets with |Aj-B. Then fis one-to-one if and only if it is onto. True O Falsearrow_forwardsuppose f: A -> B and g: B-> C are functions, and g o f: A -> C is bijection Find the inverse of the function: prove that g is surjectivearrow_forward
- Consider nonempty set A such that JA|2 2. Specify which one of the following statements is true. * P( AxA ) n P(A) =Ø There is an injective function from A xA to A. There is a bijective function from A x A onto A. O P( AXA) n P(A) = {(0}} None of the other statements is true. O AXAS P(A)arrow_forwardLet A = {1, 2, 3, 4} and B = {a, b, c}. Give an example of a function f: A -> B that is neither injective nor surjective.arrow_forwardLet f: A → B be a function with A₁, A₂ ≤ A. Determine if the following statement is true or false. Prove it, if you think it's true, or give a counterexample otherwise: ƒ(A₁ N A₂) = ƒ (A₁) Ñ ƒ(A₂). Give an example of a function from N to N that is onto but not one-to-one Let A = {1,2,3, ...,8} and consider the function f: P(A) → N given by f(B) = |B|. Prove or disprove the statement that f is one-to-one.arrow_forward
- Determine whether each of the following functions f : {a,b,c,d} -> {a,b,c,d} is one-to-one and/or onto. (a) f(a) = b, f(b) = a, f(c) = b, f(d) = c (b) f(a) = b, f(b) = b, f(c) = d, f(d) = c (c) f(a) = b, f(b) = a, f(c) = c, f(d) = d (d) f(a) = d, f(b) = a, f(c) = c, f(d) = b (e) f(a) = c, f(b) = d, f(c) = aarrow_forward9. Is the following statement TRUE or FALSE? If it is TRUE, prove it, and if it is FALSE, provide a counter-example.If f: S→T is a function, where |S| > |T| (and both are finite sets), then there exist elements s1 and s2 in S such that fs1()=fs2()arrow_forwarda) Let O be the set of odd numbers and O' = {1, 5, 9, 13, 17, ...} be its subset. Define the bijections,f and g as: f:0 0', f(d) = 2d 1, Vd e O. g: N-0, g(n) = 2n + 1, Vn e N. Using only the concept of function composition, can there be a bijective map from N to O'? If so, compute it. If not, explain in details why not.arrow_forward
- Consider sets A and B with |A| = 10 and |B| = 10 1. How many functions f: A implies B are there?arrow_forwardLet F be the set of all functions of the form f : N {0,1}. Also, let G be the set of all functions of the form f: {0, 1}* → {a, b}.Then: O (a) |F| = |G| = |N| O (b) |F| = |P(G)| 미 (c) 1지=1P(N)| □ (d) 지< IgI, but |F # I9| □ (e) |지= 19| %3D %3D Select all possible options that apply. earrow_forwardA = {0, 1, 2, 3, 4} B = {2, 3, 4, 5}Given the sets A and B, how many constants maps are there from A into B.arrow_forward
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