COMP232 - Mathematics for Computer Science Tutorial 7 Ali Moallemi moa [email protected] Iraj Hedayati h [email protected] Concordia University, Winter 2016 Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 1 / 13 Table of Contents 1 2.3 Functions Exercise 1 Exercise 2 Exercise 10 Exercise 11 Exercise 14 Exercise 15 Exercise 20 Exercise 30 Exercise 32 Exercise 40 Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 2 / 13 Exercise 1 Why is f not a function from R to R if a) f (x) = x1 Answer: f (0) is not defined √ b) f (x) = x Answer: f (x) is not defined for x < 0 p c) f (x) = ± (x2 + 1) Answer: HINT: f is a function if and only if for every input there is only one identical output. f (x) is not well-defined because there are two distinct values assigned to each x Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 3 / 13 Exercise 2 Determine whether f is a function from Z to R if a) f (n) = ±n Answer: NO for every n there are two distinct answers √ b) f (n) = n2 + 1 Answer:YES For every n in Z, n2 + 1 is positive and there is an identical answer. c) f (n) = n21−4 Answer: NO For n = ±2, the function is not defined Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 4 / 13 Exercise 10 Determine whether each of these functions from {a, b, c, d} to itself is one-to-one a) f (a) = b, f (b) = a, f (c) = c, f (d) = d Answer: YES b) f (a) = b, f (b) = b, f (c) = d, f (d) = c Answer: NO because f (a) = f (b) = b c) f (a) = d, f (b) = b, f (c) = c, f (d) = d Answer: NO because f (a) = f (d) = d Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 5 / 13 Exercise 11 Which functions in Exercise 10 are onto? a) f (a) = b, f (b) = a, f (c) = c, f (d) = d Answer: YES b) f (a) = b, f (b) = b, f (c) = d, f (d) = c Answer: NO because 6 ∃x ∈ {a, b, c, d} (f (x) = a) c) f (a) = d, f (b) = b, f (c) = c, f (d) = d Answer: NO because 6 ∃x ∈ {a, b, c, d} (f (x) = a) Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 6 / 13 Exercise 14 Determine whether f : Z × Z → Z is onto if a) f (m, n) = 2m − n Answer: YES, let m = 0 and −n will cover entire Z b) f (m, n) = m2 − n2 . Answer: NO HINT:m2 , n2 = {0, 1, 4, 9, . . .} A counterexample can be 2. 6 ∃m, n ∈ Z (m2 − n2 = 2). c) f (m, n) = m + n + 1 Answer: YES, let n = −1 and m will cover entire Z d) f (m, n) = |m| − |n|. Answer: YES if n = 0, m will cover positive members of Z if m = 0, n will cover negative members of Z e) f (m, n) = m2 − 4 Answer: NO, m2 − 4 ≥ −4 and it will not cover members of Z less than -4 Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 7 / 13 Exercise 15 Determine whether f : Z × Z → Z is onto if a) f (m, n) = m + n Answer: YES, let n = 0 and m will cover entire Z b) f (m, n) = m2 + n2 Answer: NO m2 + n2 ≥ 0 and it will not cover members of Z less than 0 c) f (m, n) = m Answer: YES d) f (m, n) = |n| Answer: NO |n| ≥ 0 and it will not cover members of Z less than 0 e) f (m, n) = m − n Answer: YES, let n = 0 and m will cover entire Z Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 8 / 13 Exercise 20 Give an example of a function from N to N that is a) one-to-one but not onto. Answer: f (n) = 2n b) onto but not one-to-one. Answer: f (n) = dn/2e c) both onto and one-to-one (but different from the identity function). ( n+1 if n is odd Answer: f (n) = n−1 if n is even d) neither one-to-one nor onto. √ Answer: f (n) = d ne + 1 Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 9 / 13 Exercise 30 Let S = {−1, 0, 2, 4, 7}. Find f (S) if a) f (x) = 1. Answer: f (S) = {1} b) f (x) = 2x + 1. Answer: f (S) = {−1, 1, 5, 9, 15} c) f (x) = dx/5e. Answer: f (S) = {0, 1, 2} d) f (x) = b(x2 + 1)/3c. Answer: f (S) = {0, 1, 5, 16} Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 10 / 13 Exercise 32 Let f (x) = 2x where the domain is the set of real numbers. What is a) f (Z). Answer: The set of all even numbers b) f (N). Answer: The set of all positive even numbers c) f (R). Answer: R Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 11 / 13 Exercise 40 Let f be a function from the set A to the set B. Let S and T be subsets of A. Show that a) f (S ∪ T ) = f (S) ∪ f (T ). Answer: ⇒) ∀y∃x y ∈ f (S ∪ T ) → x ∈ S ∪ T ∧ f (x) = y ⇒ ∀y∃x y ∈ f (S ∪ T ) → x ∈ S ∨ x ∈ T ∧ f (x) = y ⇒ ∀y y ∈ f (S ∪ T ) → y ∈ f (S) ∨ y ∈ f (T ) ⇒ ∀y y ∈ f (S ∪ T ) → y ∈ f (S) ∪ f (T ) ⇒ f (S ∪ T ) ⊆ f (S) ∪ f (T ) ⇐) (S ⊆ S ∪ T ) ∧ (T ⊆ S ∪ T ) ⇒ (f (S) ⊆ f (S ∪ T )) ∧ (f (T ) ⊆ f (S ∪ T )) ⇒ f (S) ∪ f (T ) ⊆ f (S ∪ T ) Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 12 / 13 Exercise 40 Let f be a function from the set A to the set B. Let S and T be subsets of A. Show that b) f (S ∩ T ) ⊆ f (S) ∩ f (T ). Answer: (S ∩ T ⊆ S) ∧ (S ∩ T ⊆ T ) ⇒ (f (S ∩ T ) ⊆ f (S)) ∧ (f (S ∩ T ) ⊆ f (T )) ⇒ f (S ∩ T ) ⊆ f (S) ∩ f (T ) Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 13 / 13
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