Midterm Exam PDF

Which of the following sentences is a proposition? 3 + 2 = 6 Take this pencil Can you help me? x + 2 = 6 Why should you study discrete mathematics? Let p be a proposition. The statement “It is not the case that p” is denoted by ¬𝑝𝑝 𝑝𝑝 → ¬𝑝𝑝 ⊕ 𝑝𝑝 𝑝𝑝 ∧ ¬𝑝𝑝 ¬𝑝𝑝 ∨ 𝑝𝑝 Let p and q be...

Which of the following sentences is a proposition? 3 + 2 = 6 Take this pencil Can you help me? x + 2 = 6 Why should you study discrete mathematics? Let p be a proposition. The statement “It is not the case that p” is denoted by ¬𝑝𝑝 𝑝𝑝 → ¬𝑝𝑝 ⊕ 𝑝𝑝 𝑝𝑝 ∧ ¬𝑝𝑝 ¬𝑝𝑝 ∨ 𝑝𝑝 Let p and q be propositions. The proposition that is true when both p and q are true and is false otherwise is denoted by 𝑝𝑝 ∧ 𝑞𝑞 𝑝𝑝 ∨ 𝑞𝑞 𝑝𝑝 → 𝑞𝑞 𝑝𝑝 ↔ 𝑞𝑞 𝑝𝑝 ⊕ 𝑞𝑞 Let p and q be propositions. The proposition that is false when p and q are both false and is true otherwise is denoted by 𝑝𝑝 ∨ 𝑞𝑞 𝑝𝑝 → 𝑞𝑞 𝑝𝑝 ∧ 𝑞𝑞 𝑝𝑝 ↔ 𝑞𝑞 𝑝𝑝 ⊕ 𝑞𝑞 Let p and q be propositions. The proposition that is true when exactly one of p and q is true and is false otherwise is denoted by 𝑝𝑝 ⊕ 𝑞𝑞 𝑝𝑝 ∨ 𝑞𝑞 𝑝𝑝 → 𝑞𝑞 𝑝𝑝 ∧ 𝑞𝑞 𝑝𝑝 ↔ 𝑞𝑞  Let p and q be propositions. The proposition that is false when p is true and q is false and is true otherwise is denoted by 𝑝𝑝 → 𝑞𝑞 𝑝𝑝 ∧ 𝑞𝑞 𝑝𝑝 ↔ 𝑞𝑞 𝑝𝑝 ⊕ 𝑞𝑞 𝑝𝑝 ∨ 𝑞𝑞 Let p and q be propositions. The proposition that is true when p and q have the same truth values and is false otherwise is denoted by 𝑝𝑝 ↔ 𝑞𝑞 𝑝𝑝 ⊕ 𝑞𝑞 𝑝𝑝 ∨ 𝑞𝑞 𝑝𝑝 → 𝑞𝑞 𝑝𝑝 ∧ 𝑞𝑞 Find the converse of 𝑝𝑝 → ¬𝑞𝑞 ¬𝑞𝑞 → 𝑝𝑝 𝑞𝑞 → ¬𝑝𝑝 𝑞𝑞 → 𝑝𝑝 ¬𝑞𝑞 → ¬𝑝𝑝 𝑝𝑝 → 𝑞𝑞 Find the contrapositive of ¬𝑝𝑝 → 𝑞𝑞 ¬𝑞𝑞 → 𝑝𝑝 𝑞𝑞 → ¬𝑝𝑝 𝑝𝑝 → ¬𝑞𝑞 ¬𝑞𝑞 → ¬𝑝𝑝 𝑝𝑝 → 𝑞𝑞 Find the bitwise OR of the bit strings 1011 0010 and 0110 0110 1111 0110 0010 0011 0011 0011 1010 1001 0111 1100  Find the bitwise AND of the bit strings 1010 1010 and 1001 1001 1000 1000 0101 0101 1100 1100 0011 0011 1011 1011 Find the bitwise XOR of the bit strings 0111 0101 and 1101 0101 1010 0000 1111 0111 0101 0100 0110 1010 1101 0101 Construct the truth table for the proposition (𝑝𝑝 → 𝑞𝑞) ∨ (¬𝑝𝑝 ↔ 𝑞𝑞) p q (𝑝𝑝 → 𝑞𝑞) ∨ (¬𝑝𝑝 ↔ 𝑞𝑞) T T T T F T F T T F F T p q (𝑝𝑝 → 𝑞𝑞) ∨ (¬𝑝𝑝 ↔ 𝑞𝑞) T T T T F T F T F F F T p q (𝑝𝑝 → 𝑞𝑞) ∨ (¬𝑝𝑝 ↔ 𝑞𝑞) T T T T F F F T T F F F p q (𝑝𝑝 → 𝑞𝑞) ∨ (¬𝑝𝑝 ↔ 𝑞𝑞) T T F T F F  F T T F F F p q (𝑝𝑝 → 𝑞𝑞) ∨ (¬𝑝𝑝 ↔ 𝑞𝑞) T T F T F T F T T F F F Construct the truth table for the proposition 𝑝𝑝⨁(¬𝑞𝑞 → 𝑟𝑟). p q r 𝑝𝑝⨁(¬𝑞𝑞 → 𝑟𝑟) T T T F T T F F T F T F T F F T F T T T F T F T F F T T F F F F p q r 𝑝𝑝⨁(¬𝑞𝑞 → 𝑟𝑟) T T T T T T F T T F T T T F F F F T T T F T F T F F T T F F F T p q r 𝑝𝑝⨁(¬𝑞𝑞 → 𝑟𝑟) T T T T T T F F T F T T T F F F F T T T F T F F F F T T F F F F  p q r 𝑝𝑝⨁(¬𝑞𝑞 → 𝑟𝑟) T T T T T T F T T F T T T F F T F T T T F T F T F F T T F F F F p q r 𝑝𝑝⨁(¬𝑞𝑞 → 𝑟𝑟) T T T F T T F F T F T T T F F T F T T F F T F T F F T F F F F T Let p, q and r be the propositions “You get an A on the final exam”, “You do every exercise in this book” and “You get an A in this class” respectively. Write the proposition “Getting an A on the final and doing every exercise in this book is sufficient for getting an A in this class” using p, q and r and logical connectives (𝑝𝑝 ∧ 𝑞𝑞) → 𝑟𝑟 𝑟𝑟 → (𝑝𝑝 ∧ 𝑞𝑞) (𝑝𝑝 ∨ 𝑞𝑞) → 𝑟𝑟 (𝑝𝑝 ∧ 𝑞𝑞) ↔ 𝑟𝑟 ¬𝑟𝑟 → (¬𝑝𝑝 ∧ ¬𝑞𝑞) Let p and q be the propositions “You get an A on the final exam” and “You get an A in this class” respectively. Write the proposition “To get an A in this class, it is necessary for you to get an A on the final” using p, q and logical connectives 𝑝𝑝 → 𝑞𝑞 𝑞𝑞 → 𝑝𝑝 𝑞𝑞 ↔ 𝑝𝑝 𝑝𝑝 ⨁ 𝑞𝑞 ¬𝑝𝑝 → ¬𝑞𝑞 Find the implication that is false If 2 + 3 = 5, then pigs can fly If 2 + 3 = 6, then God exists  If 2 + 3 = 4, then 3 + 3 = 5 If pigs can fly, then 1 + 3 = 5 If 2 + 3 = 5, then 1 + 3 = 4 Let p and q be the propositions “It is below freezing” and “It is snowing” respectively. Express the proposition (𝑝𝑝 ∨ 𝑞𝑞) ∧ (𝑝𝑝 → ¬𝑞𝑞) as an English sentence It is either below freezing or it is snowing, but it is not snowing if it is below freezing It is below freezing or snowing, but it is not snowing only if it is below freezing It is below freezing but not snowing If it is below freezing, then it is not snowing It is either below freezing or snowing, and it is not below freezing if it is snowing Let p and q be the propositions “You miss the final examination” and “You pass the course” respectively. Express the proposition ¬𝑝𝑝 ↔ 𝑞𝑞 as an English sentence If you don’t miss the final examination then you pass the course, and conversely That you don’t miss the final examination is necessary for passing the course Missing the final examination is sufficient for passing the course by you You don’t miss the final examination or you pass the course If you don’t pass the course, you miss the final examination A compound proposition is a tautology if it is always true, no matter what the truth values of the propositions that occur in it it is always false, no matter what the truth values of the propositions that occur in it it is always true whenever each of the propositions that occur in it is true it is only false when each of the propositions that occur in it is false it is always true whenever all the propositions that occur in it have the same truth values The propositions p and q are logically equivalent if 𝑝𝑝 ↔ 𝑞𝑞 is a tautology 𝑝𝑝 ∨ 𝑞𝑞 is a contradiction ¬( 𝑝𝑝 ∧ 𝑞𝑞) is a contradiction 𝑝𝑝 ⨁ 𝑞𝑞 is a tautology (𝑝𝑝 → 𝑞𝑞) ∧ (¬𝑞𝑞 → ¬𝑝𝑝) is a tautology Find the proposition that is a tautology ¬(𝑝𝑝 ∨ 𝑞𝑞) ↔ (¬𝑝𝑝 ∧ ¬𝑞𝑞) (𝑝𝑝 → 𝑞𝑞) ↔ (¬𝑞𝑞 ∨ 𝑝𝑝) ¬(𝑝𝑝 ∨ 𝑞𝑞) ↔ (𝑝𝑝 ∨ 𝑞𝑞) (𝑝𝑝 → 𝑞𝑞) ↔ (¬𝑞𝑞 → 𝑝𝑝) (¬𝑝𝑝 ∨ ¬𝑞𝑞) ↔ (𝑞𝑞 → 𝑝𝑝) Which of the following logical equivalences is a distributive law? 𝑝𝑝 ∧ (𝑞𝑞 ∨ 𝑟𝑟) ⇔ (𝑝𝑝 ∧ 𝑞𝑞) ∨ (𝑝𝑝 ∧ 𝑟𝑟) 𝑝𝑝 ∨ (𝑞𝑞 ∨ 𝑟𝑟) ⇔ (𝑝𝑝 ∨ 𝑞𝑞) ∨ 𝑟𝑟 𝑝𝑝 ∨ 𝑞𝑞 ⇔ 𝑞𝑞 ∨ 𝑝𝑝 𝑝𝑝 ∧ (𝑞𝑞 ∧ 𝑟𝑟) ⇔ 𝑝𝑝 ∧ (𝑞𝑞 ∧ 𝑟𝑟) 𝑝𝑝 ∧ 𝑞𝑞 ⇔ 𝑞𝑞 ∧ 𝑝𝑝 Find the proposition that is logically equivalent to 𝑝𝑝 → 𝑞𝑞 ¬𝑝𝑝 ∨ 𝑞𝑞 ¬(𝑝𝑝⨁𝑞𝑞) 𝑝𝑝 ∨ 𝑞𝑞 ¬𝑞𝑞 ∨ 𝑝𝑝 ¬( 𝑝𝑝 ∧ 𝑞𝑞) Find the proposition that is logically equivalent to ¬𝑝𝑝 ∨ ¬𝑞𝑞 ¬( 𝑝𝑝 ∧ 𝑞𝑞) ¬( 𝑝𝑝 ∨ 𝑞𝑞) (𝑝𝑝 → 𝑞𝑞) ∨ (𝑞𝑞 → 𝑝𝑝) ¬(𝑝𝑝⨁𝑞𝑞) ¬𝑝𝑝 → 𝑞𝑞 A proposition is a contingency if it is neither a tautology nor a contradiction it is both a tautology and a contradiction it is not a contradiction it is not a tautology it is not logically equivalent to any tautology Find a compound proposition involving the propositions p, q and r that is true when p and q are false and r is true, but is false otherwise ¬𝑝𝑝 ∧ ¬𝑞𝑞 ∧ 𝑟𝑟 𝑝𝑝 ∨ 𝑞𝑞 ∨ 𝑟𝑟 (¬𝑝𝑝 ∧ ¬𝑞𝑞) ∨ 𝑟𝑟 (𝑝𝑝 ∧ 𝑞𝑞) ∨ ¬𝑟𝑟 (𝑝𝑝 ↔ 𝑞𝑞) ∧ 𝑟𝑟 Find a compound proposition involving the propositions p, q and r that is false when p is false and q and r are true, but is true otherwise 𝑝𝑝 ∨ ¬𝑞𝑞 ∨ ¬𝑟𝑟 ¬𝑝𝑝 ∧ 𝑞𝑞 ∧ 𝑟𝑟 𝑝𝑝 ∧ ¬𝑞𝑞 ∧ ¬𝑟𝑟 𝑝𝑝 → ¬(𝑞𝑞 ∧ 𝑟𝑟) ¬𝑝𝑝 ∨ 𝑞𝑞 ∨ 𝑟𝑟 Find a compound proposition involving the propositions p, q and r that is true when p and q are true and r is false, but is false otherwise 𝑝𝑝 ∧ 𝑞𝑞 ∧ ¬𝑟𝑟 (𝑝𝑝 ∨ ¬𝑟𝑟) ∧ (𝑞𝑞 ∨ ¬𝑟𝑟) (𝑝𝑝 ∨ 𝑞𝑞) ∧ ¬𝑟𝑟 𝑟𝑟 → ¬(𝑝𝑝 ∧ 𝑞𝑞) (𝑝𝑝 ∧ 𝑞𝑞) → 𝑟𝑟 Let P(x) be the statement “x spends less than three hours every weekday in class”, where the universe of discourse for x is the set of students. Express the proposition “∃𝑥𝑥¬𝑃𝑃(𝑥𝑥)” in English There is a student who spends no less than three hours every weekday in class Every student doesn’t spend more than three hours every weekday in class There is a student who doesn’t spend more than three hours every weekday in class There is a student who spends less than two hours every weekday in class Every student spends no more than six hours every weekday in class Let P(x, y) be the statement “x has taken y”, where the universe of discourse for x is the set of all students in your class and for y is the set of all computer courses at your school. Express the proposition ∃𝑦𝑦∀𝑥𝑥𝑥𝑥(𝑥𝑥, 𝑦𝑦) in English There is a computer course at your school taken by all students in your class There is a computer course at your school taken by at least one student in your class For every student in your class there is a computer course at your school taken by him (or her). For each of the computer courses at your school there is a student at your class who has taken it There is a student at your class who has taken all computer courses at your school Let P(x) be the statement “x can speak Kazakh” and let Q(x) be the statement “x knows the computer language Delphi”, where the universe of discourse for x is the set of all students at your university. Express the sentence “There is a student at your university who can speak Kazakh but who doesn’t know Delphi” in terms of P(x), Q(x), quantifiers and logical connectives. ∃𝑥𝑥(𝑃𝑃(𝑥𝑥) ∧ ¬𝑄𝑄(𝑥𝑥)) ∃𝑥𝑥(𝑃𝑃(𝑥𝑥) ∨ ¬𝑄𝑄(𝑥𝑥)) ∀𝑥𝑥(𝑃𝑃(𝑥𝑥) → ¬𝑄𝑄(𝑥𝑥)) ∃𝑥𝑥(𝑄𝑄(𝑥𝑥) → 𝑃𝑃(𝑥𝑥)) ∀𝑥𝑥(¬𝑃𝑃(𝑥𝑥) ∨ 𝑄𝑄(𝑥𝑥)) Let S(x, y) be the statement “𝑥𝑥 + 𝑦𝑦 = 𝑥𝑥 ∙ 𝑦𝑦”, where the universe of discourse for both variables is the set of integers. Which of the following statements is true? ∃𝑥𝑥∃𝑦𝑦𝑦𝑦(𝑥𝑥, 𝑦𝑦) ∀𝑥𝑥∀𝑦𝑦¬𝑆𝑆(𝑥𝑥, 𝑦𝑦) ∀𝑥𝑥∃𝑦𝑦𝑦𝑦(𝑥𝑥, 𝑦𝑦) ∀𝑦𝑦∃𝑥𝑥𝑥𝑥(𝑥𝑥, 𝑦𝑦) ∃𝑦𝑦∀𝑥𝑥𝑥𝑥(𝑥𝑥, 𝑦𝑦) Let S(x, y) be the statement “x + 3y = 3x – y”, where the universe of discourse for both variables is the set of integers. Which of the following statements is true? ∀𝑦𝑦∃𝑥𝑥𝑥𝑥(𝑥𝑥, 𝑦𝑦) ∀𝑥𝑥∀𝑦𝑦¬𝑆𝑆(𝑥𝑥, 𝑦𝑦) ∀𝑥𝑥∃𝑦𝑦𝑦𝑦(𝑥𝑥, 𝑦𝑦) ∃𝑥𝑥∀𝑦𝑦𝑦𝑦(𝑥𝑥, 𝑦𝑦) ∃𝑦𝑦∀𝑥𝑥𝑥𝑥(𝑥𝑥, 𝑦𝑦) Rewrite the statement ¬∃𝑦𝑦∀𝑥𝑥𝑥𝑥(𝑥𝑥, 𝑦𝑦) so that negations appear only within predicates (that is, so that no negation is outside a quantifier or an expression involving logical connectives) ∀𝑦𝑦∃𝑥𝑥¬𝑃𝑃(𝑥𝑥, 𝑦𝑦) ∀𝑦𝑦∀𝑥𝑥𝑥𝑥(𝑥𝑥, 𝑦𝑦) ∃𝑦𝑦∀𝑥𝑥¬𝑃𝑃(𝑥𝑥, 𝑦𝑦) ∃𝑥𝑥∀𝑦𝑦¬𝑃𝑃(𝑥𝑥, 𝑦𝑦) ∀𝑥𝑥∃𝑦𝑦𝑦𝑦(𝑥𝑥, 𝑦𝑦) Which of the following statements is true if the universe of discourse for all variables is the set of all integers? ∀𝑛𝑛∃𝑚𝑚(𝑛𝑛2 + 1 < 𝑚𝑚 − 1) ∃𝑛𝑛(𝑛𝑛2 = 8) ∀n(n2 ≥1) ∀𝑛𝑛(𝑛𝑛2 ≥ 1) ∃𝑛𝑛∀𝑚𝑚(𝑛𝑛 < 𝑚𝑚3 ) ∀𝑛𝑛∀𝑚𝑚(𝑛𝑛 > 𝑚𝑚 ∨ 𝑛𝑛2 ≤ 𝑚𝑚2 )  Which of the following statements is true if the universe of discourse of each variable is the set of real numbers? ∃𝑥𝑥(𝑥𝑥2 = 8) ∀𝑥𝑥∃𝑦𝑦(𝑥𝑥 = 𝑦𝑦2 ) ∃𝑥𝑥∀𝑦𝑦(𝑥𝑥 ∙ 𝑦𝑦 = 4) ∀𝑥𝑥∃𝑦𝑦(𝑥𝑥 + 𝑦𝑦 ≠ 𝑦𝑦 + 𝑥𝑥) ∃𝑦𝑦∀𝑥𝑥(𝑥𝑥3 = 𝑦𝑦) When the statement ∀𝑦𝑦∃𝑥𝑥𝑥𝑥(𝑥𝑥, 𝑦𝑦) is false? There is a y such that P(x, y) is false for every x For every y there is an x for which P(x, y) is true There is an x such that P(x, y) is false for every y There is a pair x, y for which P(x, y) is false For every y there is an x for which P(x, y) is false When the statement ∃𝑦𝑦∀𝑥𝑥𝑥𝑥(𝑥𝑥, 𝑦𝑦) is true? There is a y for which P(x, y) is true for every x For every y there is an x for which P(x, y) is true For every x there is a y for which P(x, y) is false There is a pair x, y for which P(x, y) is true There is an x for which P(x, y) is true for every y List the members of the set {x | x is a negative integer greater than (– 8)} {– 7, – 6, – 5, – 4, – 3, – 2, –1} {– 8, –7, – 6, – 5, – 4, – 3, – 2, –1} {–9, –10, –11, …} {0, 1, 2, 3, 4, 5, 6, 7} {9, 10, 11, 12, …} Find the power set of {0, 1}. {∅, {0}, {1}, {0, 1}} {∅, {0, 1}} {{0}, {1}, {0, 1}} {{0}, {1}} {∅, {1}} Let A = {1, 2, 3, 4, 5}, B = {2, 4, 6, 8} and C = {2, 4}. Which of the following statements is true?  𝐴𝐴 ∩ 𝐵𝐵 ⊆ 𝐶𝐶 𝐴𝐴 ⊆ 𝐵𝐵 ∪ 𝐶𝐶 𝐴𝐴 − 𝐶𝐶 ⊆ 𝐵𝐵 𝐵𝐵 − 𝐶𝐶 ⊆ 𝐴𝐴 𝐵𝐵 ⊆ 𝐴𝐴 ∩ 𝐶𝐶 Let A = {a, b, c, d, e, f}, B = {b, d, f, g, h, s} and C = {m, n, o, p, r, d, f}. Find (𝐴𝐴 ∩ 𝐶𝐶) ∪ 𝐵𝐵 {b, d, f, g, h, s} {b, m, n, o, p, r, d , f} {a, b, c, d, e, f} {m, n, o, p, r, d, f} {b, d, e, f, g, h, r} Let A = {0, 1, 2}, B = {y, z} and C = {b, c}. Find 𝐶𝐶 × 𝐵𝐵 × 𝐶𝐶 {(b, y, b), (b, y, c), (b, z, b), (b, z, c), (c, y, b), (c, y, c), (c, z, b), (c, z, c)} {(0, y, b), (1, y, b), (2, y, b), (0, z, b), (1, z, c), (2, z, c), (0, y, c), (1, y, c), (2, y, c)} {(b, y),(b, z), (c, y), (c, z)} {(b, 0), (b, 1), (b, 2), (c, 0), (c, 1), (c, 2), (0, y), (1, y), (2, y)} {(b, y, c), (c, y, b), (b, z, c), (c, z, b)} A set A is a proper subset of a set B if every element of A is also an element of B and 𝐴𝐴 ≠ 𝐵𝐵 every element of A is also an element of B there is an element of B that is not an element of A there is an element of A that is also an element of B 𝐵𝐵 – 𝐴𝐴 ≠ ∅ and 𝐴𝐴 ∩ 𝐵𝐵 ≠ ∅ Let A be a set. The power set of A is the set of all subsets of A the set of all proper subsets of A {∅, 𝐴𝐴} the set of all finite subsets of A the set of all proper subsets of A and the empty set Let A and B be sets. The Cartesian product of A and B is the set of all ordered pairs (a, b) where 𝑎𝑎 ∈ 𝐴𝐴 𝑎𝑎𝑎𝑎𝑎𝑎 𝑏𝑏 ∈ 𝐵𝐵 {(𝑎𝑎, 𝑏𝑏)|(𝑎𝑎 ∈ 𝐴𝐴 ∧ 𝑏𝑏 ∉ 𝐵𝐵) ∨ (𝑎𝑎 ∉ 𝐴𝐴 ∧ 𝑏𝑏 ∈ 𝐵𝐵)}  the set of all ordered pairs (b, a) where 𝑎𝑎 ∈ 𝐴𝐴 𝑎𝑎𝑎𝑎𝑎𝑎 𝑏𝑏 ∈ 𝐵𝐵 the set of all two-element sets {a, b} where 𝑎𝑎 ∈ 𝐴𝐴 𝑎𝑎𝑎𝑎𝑎𝑎 𝑏𝑏 ∈ 𝐵𝐵 Let A and B be sets. The union of A and B is the set that contains those elements that are either in A or in B, or in both {(𝑎𝑎, 𝑏𝑏)|(𝑎𝑎 ∈ 𝐴𝐴 ∧ 𝑏𝑏 ∉ 𝐵𝐵) ∨ (𝑎𝑎 ∉ 𝐴𝐴 ∧ 𝑏𝑏 ∈ 𝐵𝐵)} the set that contains those elements that are only either in A or in B the set that contains those elements that are in both A and B the set that contains those elements that are in A but not in B Let A and B be sets. The intersection of A and B is the set containing those elements that are in both A and B the set that contains those elements that are either in A or in B, or in both. {𝑥𝑥|(𝑥𝑥 ∈ 𝐴𝐴) ∨ (𝑥𝑥 ∈ 𝐵𝐵)} the set that contains those elements that are in A but not in B the set that contains those elements that are only either in A or in B Let U = {1, 2, 3, 4, 5, 6, 7, 8} be the universal set, and A = {1, 3, 4, 5, 8}, B = {2, 4, 5, 7, 8}. Find the complement of A {2, 6, 7} {1, 3, 6} {4, 5, 8} {6} {1, 2, 3, 4, 5, 7, 8} Let f be a function from A to B. Then the codomain of f is the set B the set A {𝑏𝑏 ∈ 𝐵𝐵 | 𝑡𝑡ℎ𝑒𝑒𝑒𝑒𝑒𝑒 𝑖𝑖𝑖𝑖 𝑎𝑎𝑎𝑎 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑎𝑎 ∈ 𝐴𝐴 𝑠𝑠𝑠𝑠𝑠𝑠ℎ 𝑡𝑡ℎ𝑎𝑎𝑎𝑎 𝑓𝑓(𝑎𝑎) = 𝑏𝑏} { 𝑎𝑎 ∈ 𝐴𝐴 | 𝑡𝑡ℎ𝑒𝑒𝑒𝑒𝑒𝑒 𝑖𝑖𝑖𝑖 𝑎𝑎𝑎𝑎 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑏𝑏 ∈ 𝐵𝐵 𝑠𝑠𝑠𝑠𝑠𝑠ℎ 𝑡𝑡ℎ𝑎𝑎𝑎𝑎 𝑓𝑓(𝑎𝑎) = 𝑏𝑏} {𝑏𝑏 ∈ 𝐵𝐵 | 𝑡𝑡ℎ𝑒𝑒𝑒𝑒𝑒𝑒 𝑖𝑖𝑖𝑖 𝑛𝑛𝑛𝑛 𝑎𝑎 ∈ 𝐴𝐴 𝑤𝑤𝑤𝑤𝑤𝑤ℎ 𝑓𝑓(𝑎𝑎) = 𝑏𝑏} Let f be the function that assigns the first three bits of a bit string of length 3 or greater to that string. Then the codomain of f is {000, 001, 010, 100, 011, 101, 110, 111} {00, 01, 10, 11} the set of all bit strings of length 3 or greater {0, 1} ∅ Let A = {a, b, c, d, e, g, h} and B = {0, 1, 3, 4, 5} with f(a) = 3, f(b) = 2, f(c) = 4, f(d) = 0, f(e) = 5, f(g) = 1 and f(h) = 3. Find the image of S = {c, d, e, g}. {0, 1, 4, 5} {4, 0, 3, 1} {0, 1, 2, 3, 4, 5} {4, 0, 5, 1, 3} {1, 2, 3, 4} A function f: 𝐴𝐴 → 𝐵𝐵 is said to be injective … iff f(x) = f(y) implies that x = y for all x and y in the domain of f iff for every element 𝑏𝑏 ∈ 𝐵𝐵 there is an element 𝑎𝑎 ∈ 𝐴𝐴 with f(a) = b if f(x) < f(y) whenever x < y and x and y are in the domain of f if 𝑓𝑓(𝑥𝑥) ≠ 𝑓𝑓(𝑦𝑦) implies that 𝑥𝑥 ≠ 𝑦𝑦 for all x and y in the domain of f iff for every element 𝑎𝑎 ∈ 𝐴𝐴 there is an element 𝑏𝑏 ∈ 𝐵𝐵 with f(a) = b A function f: 𝐴𝐴 → 𝐵𝐵 is said to be surjective … iff for every element 𝑏𝑏 ∈ 𝐵𝐵 there is an element 𝑎𝑎 ∈ 𝐴𝐴 with f(a) = b iff for every element 𝑎𝑎 ∈ 𝐴𝐴 there is an element 𝑏𝑏 ∈ 𝐵𝐵 with f(a) = b if f(x) < f(y) whenever x < y and x and y are in the domain of f if f(x) > f(y) whenever x < y and x and y are in the domain of f iff f(x) = f(y) implies that x = y for all x and y in the domain of f The function f is a one-to-one correspondence, or a bijection, if it is both one-to-one and onto it is either one-to-one or onto it is one-to-one but not onto it is onto but not one-to-one the codomain and the range of f coincide Let f and g be the functions from the set of integers to the set of integers defined by f(x) = 3x – 4 and g(x) = 4x – 3. Find the composition of g and f 12x – 19 12x – 13 7x – 7 7x – 12 x + 1  Let f be the function that assigns to each positive integer its first digit. Find the range of f {1, 2, 3, 4, 5, 6, 7, 8, 9} {𝑥𝑥 ∈ 𝑍𝑍|𝑥𝑥 > 0} {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} {x | x is an integer and x ≥ 0} the set of integers 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 ↳ 1,98 ↲. 1 1,98 2 0 1,99 𝑭𝑭𝑭𝑭𝑭𝑭𝑭𝑭 ↱ 𝟏𝟏, 𝟎𝟎𝟎𝟎 ↰ 2 1 1,01 0 1,02 Find ↳– 𝟐𝟐, 𝟑𝟑𝟑𝟑 ↲ -3 -2,34 0 -2 -1 Find ↱– 𝟐𝟐, 𝟑𝟑𝟑𝟑 ↰. -2 -3 -2,34 0 -1 Find ↳ 𝟏𝟏/𝟏𝟏𝟏𝟏 + ↱ 𝟗𝟗/𝟏𝟏𝟏𝟏 ↰↲. 1 9/10 0 1/10 2 Find ↱ 𝟏𝟏/𝟏𝟏𝟏𝟏 +↳ 𝟗𝟗/𝟏𝟏𝟏𝟏 ↲↰. 1 2 9/10 1/10 0 Find ↱↳ 𝟏𝟏/𝟑𝟑 ↲ + ↱ 𝟏𝟏/𝟑𝟑 ↰ +𝟏𝟏/𝟑𝟑 ↰. 2 1 0 1/3 2/3 Which of the following functions from {a, b, c, d} to itself is one-to-one? f(a) = d, f(b) = c, f(c) = b, f(d) = a f(a) = c, f(b) = d, f(c) = b, f(d) = b f(a) = a, f(b) = b, f(c) = d, f(d) = d f(a) = a, f(b) = a, f(c) = a, f(d) = a f(a) = d, f(b) = a, f(c) = a, f(d) = d 𝐿𝐿𝐿𝐿𝐿𝐿 𝑆𝑆 = {– 3, 0, 3, 7}. 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝑓𝑓(𝑆𝑆) 𝑖𝑖𝑖𝑖 𝑓𝑓(𝑥𝑥) = ↱ ( 𝑥𝑥 2 + 2)/3 ↰. {1, 4, 17} {0, 3, 17} {–2, 0, 3, 16} {–3, 0, 3, 7} Find ↱ 𝟐𝟐/𝟑𝟑 – ↳ 𝟒𝟒/𝟑𝟑 ↲↰ 0 -2/3 2 1 3