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Discrete Math Final
| Question | Answer |
|---|---|
| In a college, 200 students are randomly selected. 140 like tea, 120 like coffee and 80 like both. How many students like neither tea nor coffee? The given information may be represented by the following Venn diagram, where T = tea and C = coffee. | 20 |
| Consider the above diagram where the number in each region indicates how many elements are there in that region. Then what is the number of elements in set A? | 14 |
| The Venn diagram below shows the number of girls on the soccer and track teams at a high school. How many girls are both on the track teams? | 37 |
| Consider the above diagram where the number in each region indicates how many elements are there in that region. Then what is THE NUMBER OF ELEMENTS IN SETS A and C? | 6 |
| The Venn diagram below shows the number of girls on the soccer and track teams at a high school. How many girls are at the school in all? | 5 |
| GIVEN: U = {1,2,3,4,5} A = {1,2,3} B = {5} Find 𝐴∩𝐵? | {∅} |
| Given: U = {1,2,3,4,5} A = {1,2,3} B = {5} Find 𝐴′∪𝐵? | {4,5} |
| Given: U = {1,2,3,4,5} A = {1,2,3} B = {5} Find 𝐴′∩𝐵 ? | {5} |
| Given: U = {1,2,3,4,5} A = {1,2,3} B = {5} Find 𝐴∪𝐵? | {1,2,3,5} |
| Given: U = {a,b,c,d,e} A = {c,d,e} B = {a,c,d} Find 𝐴′∪𝐵? | {a,b,c,d,} |
| The compound propositions p and q are called logically equivalent if ________ is a tautology. a) p ↔ q b) p → q c) ¬ (p ∨ q) d) ¬p ∨ ¬q | A) p ↔ q |
| ¬ (p ↔ q) is logically equivalent to: a) p ↔ ¬q b) ¬p ↔ q c) ¬p ↔ ¬q d) ¬q ↔ ¬p | a) p ↔ ¬q |
| Suppose p is false, q is false, s is true. Find the truth value of (s∨p)∧(q∧~s) | False |
| Select the statement that is the negation of "Today is Monday and it isn't raining." A. Today isn't Monday and it isn't raining. B. Today isn't Monday or it isn't raining. C. Today isn't Monday or it is raining. | C. Today isn't Monday or it is raining. |
| D. Today isn't Monday and it is raining. E. Today is Friday and it is snowing. | C. Today isn't Monday or it is raining. |
| If A is any statement, then which of the following is a tautology? a) A ∧ F b) A ∨ F c) A ∨ ¬A d) A ∧ T | c) A ∨ ¬A |
| compound proposition that is neither a tautology nor a contradiction is called a ___________ a) Contingency b) Equivalence c) Condition d) Inference | a) Contingency |
| It is a declarative sentence that is either true or false. | proposition |
| The truth value of given statement is ‘If 9 is prime then 3 is even’. a) False b) True | b) True |
| If two angles are supplementary, then they have a sum of 180. Is the inverse true? | A. Yes, if two angles are not supplementary, then they do not add to 180 |
| Conditional: If it does not rain, then we will have practice. What is "If it rains today, then we will have not practice", called | inverse |
| ”Everyone wants to learn cosmology.” This argument may be true for which domains? | c) Both of the mentioned |
| Let R (x) denote the statement “x > 2.” What is the truth value of the quantification ∃xR(x), having domain as real numbers? | a) True |
| Determine the truth value of statement ∃n (4n = 3n) if the domain consists of all integers. | a) True |
| The statement, “At least one of your friends is perfect”. Let P (x) be “x is perfect” and let F (x) be “x is your friend” and let the domain be all people. | c) ∃x (F (x) ∧ P (x)) |
| ∀x∃yP (x, y) is equivalent to ∃y∀xP (x, y). | False |
| Symbol ∀ denotes “for all” is called | universal quantifier |
| There are __________________ pens in that drawer. | some |
| A rule of inference that introduces existential quantifiers. | Existential Generalization |
| The quantifier used to translate particular statements in predicate logic | Existential Quantifier |
| This rule says if P is true for every element of the domain, we can name such an element by an arbitrary variable name like x, y, or z, or we can specify a particular constant in the domain, and P is still true for all these things. | Universal Instantiation |
| All flowers are plants. Sunflower is a flower. Therefore, sunflower is a plant. | Universal Instantiation |
| – P(x) is “xisaplant” – a is a constant symbol (Sunflower) – F(x) is “x is a flower” – The argument is ( x)[F(x) P(x)] Λ F(a) P(a) • The proof sequence is as follows: | Universal Instantiation |
| 1. ( x)[F(x) P(x)] - hyp. 2. F (a) - hyp 3. F(a) P(a - 1, 4. P(a) - 2,3 , mp | Universal Instantiation |
| All racers live dangerously. Gomer is a racer. Therefore, Gomer lives dangerously. | True |
| Fido is barking. Someone must be in the front yard. | The only time Fido barks is when someone is in the front yard. |
| is this statement is true? P ⇒ Q. Proof. Assume, for the sake of contradiction P is true but Q is false. ··· Since we have a contradiction, it must be that Q is true. | True |
| In proving √5 as irrational, we begin with assumption √5 is rational in which type of proof? | Proof by contradiction |
| If m and n are non-zero integers, is (m+n)/mn a rational? | True |
| Consider the statement, “If n is divisible by 30 then n is divisible by 2 and by 3 and by 5.” Which of the following statements is equivalent to this statement? | If n is not divisible by 2 or not divisible by 3 or not divisible by 5 then n is not divisible by 30. |
| An example that shows a statement is false. | counterexample |
| Let n and m be integers, then if one of n and m is even and the other is odd, then n+m is _________________. | odd |
| The statement below is an example of what type of rule of inference. P → Q means“if there is a storm, then the office is closed.” Q → R means“if the office is closed, then I don’t go to work.” P → R means“if there is a storm, then I don’t go to work.” | Hypothetical Syllogism |
| Decide if the following arguments are valid or invalid. State the Rule of Inference of fallacy used If the movie is long, I will fall asleep. I do fall asleep. Therefore the movie was long. | invalid |
| __________ bytes are required to encode 2000 bits of data. | b) 2 |
| The function f(x) = x3 is bijection from R to R. Is it True or False? | a) True |
| The little-o notation for f(x) = xlogx is | c) x2 |
| The big-Omega notation for f(x) = 2x4 + x2 – 4 is | d) x4 |
| State whether the given statement is true or false The range of function f(x) = sin(x) is (-∞, ∞). | b) False |
| State whether the given statement is true or false Codomain is the subset of range. | b) False |
| A ceil function map a real number to : | c) smallest following integer |
| Floor(2.4) + Ceil(2.9) is equal to | c) 5 |
| Set A has 3 elements and set B has 4 elements then number of injections defined from A to B are? | b) 24 |
| A mapping f : X -> Y is one one if : | b) If f(x1) = f(x2) then x1 = x2 for all x1, x2 in X |
| How many numbers of three digits can be formed with digits 1, 3, 5, 7 and 9? | B) 125 |
| 14 different letters of alphabet are given, words with 6 letters are formed from these given letters. How many number of words are there which have at least one letter repeated? | b) 999988 |
| Find the number of ways in which 4 people E, F, G, H, A, C can be seated at a round table, such that E and F must always sit together. | d) 48 |
| The code for a safe is of the form PPPQQQQ where P is any number from 0 to 9 and Q represents the letters of the alphabet. How many codes are possible for each of the following cases? Note that the digits and letters of the alphabet can be repeated. | d) 456976000 |
| There are two different Geography books, five different Natural Sciences books, three different History books and four different Mathematics books on a shelf. | c) 829440 |
| How many strings of 5 letters contain ’a’ at least once? | 2115751 |
| For her English literature course, Ruchika has to choose one novel to study from a list of ten, one poem from a list of fifteen and one short story from a list of seven. How many different choices does Rachel have? | d) 10500 |
| How many even 4 digit whole numbers are there? | c) 4500 |
| In a computer science department, a student club can be formed with either 10 members from first year or 8 members from second year or 6 from third year or 4 from final year. | 25 |
| In a group of 267 people how many friends are there who have an identical number of friends in that group? | b) 2 |
| What is a full binary tree? | a) Each node has exactly zero or two children |
| The number of edges from the node to the deepest leaf is called _________ of the tree. | a) Height |
| A polytree is called _______________ | a) directed acyclic graph |
| What is a complete binary tree? | c) A binary tree, which is completely filled, with the possible exception of the bottom level, which is filled from left to right |
| What is the degree of vertex E in the given graph? | C. 6 |
| For the given graph, which of the following is true regarding its Euler properties? | Euler path and Euler circuit |
| Refer to the above graph and choose the best answer. | Hamiltonian path |
| Who introduced the concept and formulated the first approach for solving the Hamiltonian Path problem? | a) William Rowan Hamilton |
| Which of the following is true? a) A graph may contain no edges and many vertices b) A graph may contain many edges and no vertices | b) A graph may contain many edges and no vertices |
| In a simple graph, the number of edges is equal to twice the sum of the degrees of the vertices. | False |
| Algebra of logic is termed as ______________ | c) Arithmetic logic |
| A ________ value is represented by a Boolean expression. | d) Boolean |
| The ___________ of all the variables in direct or complemented from is a maxterm. | a) addition |
| Which law says 'Break the line, change the sign'? | a. De Morgans |
| This declares you can multiply or factor out an expression which is what you do in ordinary algebra | a De Morgans |
| How many NAND gates are required to make an XOR gate? | c) 4 |
| The logic gate that provides high output for same inputs ____________ | b) X-NOR |
| What is the output of an AND gate if its inputs are 1 and 0? | 0 |
| _________ is used to implement the Boolean functions. | c) Logic gates |
| WHAT IS THE OUTPUT OF AN AND GATE IF ITS INPUTS ARE 1 AND 1. | 1 |
| A NOR gate can be derived from ______ | a) NAND gate |
| Using which component a shift register is implemented? | a) register |
| What is the output of an OR gate if its inputs are 0 and 0? | 0 |
| Find the simplified expression A’BC’+AC’. | c) (A+B)C’ |
| If an expression is given that x+x’y’z=x+y’z, find the minimal expression of the function F(x,y,z) = x+x’y’z+yz? | c) x + z |
| What is the simplification value of MN(M + N’) + M(N + N’)? | b) MN+M’N’ |
| If an expression is given that x+x’y’z=x+y’z, find the minimal expression of the function F(x,y,z) = x+x’y’z+yz? | c) x + z |
| There are _________ numbers of Boolean functions of degree n. | b) 2(2*n) |
| If there are ‘M’ switches in parallel numbered from 1, 2, …, M. For circuit to be complete and bulb to glow which of the following is necessary | a) 1∧ 2∧ 3 ∧ … ∧M should be on |
| If there are ‘M’ switches in series numbered from 1, 2, …, M. For circuit to be complete and bulb to glow which of the following is necessary | a) 1∧ 2∧ 3 ∧ … ∧M should be on |
| a? is equivalent to | c) a+ϵ |
| a+b)* is equivalent to | b) (a*b*)* |
| A language is regular if and only if | a) accepted by DFA |
| Precedence of regular expression in decreasing order is | a) * , . , + |
| Regular expression are | a) Type 0 language |
| Regular expression Φ* is equivalent to | a) ϵ |
| In lexical analysis of a compiler______ is used. | a) DFA |
| A deterministic automaton system can have ______ transition for a given state of an input symbol. | a) exactly one |
| Which of the following is not a member of the set of a deterministic finite state machine? | b) initial state |
| Equivalence of automata states that ____________ | a) two automata accept the same set of input strings |
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