Solved: By constructing truth tables, decide which of the following are tautologies. (I) ~(P ^ ~P) (II) P implies ~P (III) (P ^ (p implies q)) implies q B) show that (P implies Q) implies R is logically equivalent to (~ P implies R) ^ (Q implies R)

1. Consider the K-Maps given below. For each K- Map i. Write the appropriate standard form (SOP/POS) of Boolean expression. ii. Design the circuit using AND, NOT and OR gates. iii. Design the circuit only by using • NAND gates i…

We have 5 distinct jobs to be finished in the first 22 days of June but no two of the jobs will be finished on consecutive days. In how many ways can we plan the finishing days for these 5 distinct jobs? Write your answer: the number …

Define a binary relation P from R to R as follows: for all real numbers x and y, (x,y)∈P⇔x=y^2. Is P a function? Explain.

given that the function f(x) = 4x + 1 , find a formula for f-1(x)

Q1: Determine whether the relation R on the set of all people is reflexive, symmetric, antisymmetric, and/or transitive, where (a,b) ∈ R if and only if a) a is taller than b. b) a and b were born on the same day. c) a has the sam…

Q2: Let R be the parent relation on the set of all people (see Example 21 in section 9.1 of the book). When is an ordered pair in the relation R^3? SUGGESTED TEXT: · Keneth H. Rosen. Discrete Mathematics and its Applications…

Q3: List the 16 different relations on the set {0,1}. Note: No partial credit would be admissible in this question.

(i) Use Karnaugh map to minimize the following SOP expression and also implement the simplified expression. Y = A’BC + A’B’C’ + ABC’ + AB’C’

A class contains 10 students with 6 boys and 4 girls. Find the number of ways: i. A 4 member committee can be selected from the students. ii. A 4 member committee with 2 boys and 2 girls can be selected from the students. iii. A 4 mem…

draw a binary search tree by inserting the values 50, 76, 21, 4, 32, 64, 15, 52, 14, 100, 83, 2, 3 and 70.

There were 100 students in the library who responded to how they completed their research paper. • 18 students only used the periodicals. • 29 students used the web and books. • 15 students used books, the web, and periodicals. • 40 s…

There are 70 women in class. Each play at least one game of the following games; volleyball, basketball, table tennis. 20 play volleyball only,10 play basketball only and 6 play table tennis. 4 play all the games and an equal number…

Give an example of a function which represents all types of a function. Find the composite function (f o g) (x) given that f = {(1,6), (4,7), (5,0)} and g = {(6,1), (7,4), (0,5)}

There were 100 students in the library who responded to how they completed their research papers. • 18 students only used the periodicals. • 29 students used the web and books. • 15 students used books, the web, and periodicals. •…

Let A = {a,b,c}, B = {x,y}, and C = {0,1}. Find a) A×B×C.

a) thesetofpeoplewhospeakEnglish,thesetofpeople who speak English with an Australian accent

2 Draw a Venn diagram of set A, B, C where (i) A and B have elements in common, B and C have element in common but A and C are disjoint.

Check whether the relation R ={(x, y)∈N×N | xy is the square of an integer} is an equivalence relation on N.

There are 70 women in class. Each play at least one game of the following games; volleyball, basketball, table tennis. 20 play volleyball only,10 play basketball only and 6 play table tennis. 4 play all the games and an equal number…

Suppose f:X→Y and g:Y→Z and both of these are one-to-one and onto. Prove that (g∘f)^(-1) exists and that (g∘f)^(-1)=f^(-1)∘g^(-1).

Write NEGATION of the following statements (Careful, NEGATION is not the same as INVERSE) a)If P is a square, then P is a rectangle. b)If n is prime, then n i odd or n is 2. c)If Aangloo is Meena's father, then Baangloo is her uncl…

How many strings of length 5 can be made using the letters " ABCDE" which start with A and in which CD always comes together? 1 * 3!*2!

Let p, q, r and s represent the following: Determine the truth value of each statement.P: 2 is the smallest prime number. Trueq: The capital city of Camiguin is Mambajao. Truer: The sum of 2 odd numbers is odd. Falses: The diagonals o…

Let p, q, r and s represent the following: Determine the truth value of each statement. P: 2 is the smallest prime number. True q: The capital city of Camiguin is Mambajao. True r: The sum of 2 odd numbers is odd. False s…

MODULE 6: Draw the logic diagram of the function and find for the simplified Boolean expression of the following: 1. F(a, b, c) = ∑(0, 2, 3, 4, 5, 6) 2. F(a, b, c, d) = ∑(0, 2, 3, 4, 6, 8, 10, 12, 13, 14, 15) 3. F(a, b, c, d) =…

How many distinguishable permutations with repeated elements are there for the number 20366650?

Let R be a relation on the set of all non-negative integers defined by aRb if and only if a3 - b3 is divisible by 6. Then

Let X = {2, 3, 6, 12, 24, 36}and relation  be such that "x  y" if x divides y then draw a Hasse diagram of (X, ).

Write CONVERSE of the following statements. a)If P is a square, then P is a rectangle. b)If n is prime, then n is odd or n is 2. c)If Aangloo is Meena's father, then Baangloo is her uncle and Bingli is her Aunt. d)A positive integ…

Draw the logic diagram of the function and find for the simplified Boolean expression of the following: 1. F(a, b, c) = ∑(0, 2, 3, 4, 5, 6) 2. F(a, b, c, d) = ∑(0, 2, 3, 4, 6, 8, 10, 12, 13, 14, 15) 3. F(a, b, c, d) = ∏(1, 2, 4, 5…

Write CONTRPOSITIVE of the following statements. a)If P is a square, then P is a rectangle. b)If n is prime, then n is odd or n is 2. c)If Aangloo is Meena's father, then Baangloo is her uncle and Bingli is her Aunt. d)A positive …

The police have three suspects for the murder of Janab Chaudary AllahDitta (Ch AD): Mr Chaloosak,Mr Maloosak and Mr Kabacha. Chaloosak, Maloosak and Kabacha each declare that they did not kill Ch AD. Chaloosak also states that Ch AD…

2 Draw a Venn diagram of set A, B, C where (i) A and B have elements in common, B and C have element in common but A and C are disjoint.

Check whether the relation R ={(x, y)∈N×N | xy is the square of an integer} is an equivalence relation on N.

Let p, q, r and s represent the following: Determine the truth value of each statement. P: 2 is the smallest prime number. True q: The capital city of Camiguin is Mambajao. True r: The sum of 2 odd numbers is odd. False s…

MODULE 6: Draw the logic diagram of the function and find for the simplified Boolean expression of the following: 1. F(a, b, c) = ∑(0, 2, 3, 4, 5, 6) 2. F(a, b, c, d) = ∑(0, 2, 3, 4, 6, 8, 10, 12, 13, 14, 15) 3. F(a, b, c, d) =…

How many distinguishable permutations with repeated elements are there for the number 20366650?

Let R be a relation on the set of all non-negative integers defined by aRb if and only if a3 - b3 is divisible by 6. Then

Let X = {2, 3, 6, 12, 24, 36}and relation  be such that "x  y" if x divides y then draw a Hasse diagram of (X, ).

inductive reasoning or deductive reasoning: • 17 and 5 are prime numbers, so all odd numbers are prime numbers

12 Question 3: Mathematical proficiency and the construction of mathematics ideas. To answer this question, you need to complete the table below, and also study paragraphs 2.7-2.13 in the study guide: the most important is that yo…

Un+2-5Un+1-6Un=4^n+n

Define intersection of two Fuzzy set with example.

Mary Ann pays a monthly fee for her cell phone package which includes 700 minutes. She gets Billed an additional charge for every minute she uses the phone past the 700 minutes. During her first month, Mary Ann used 26 additional minu…

Let R be the relation on Z (the set of integers) defined by (x, y)  R iff x 2 + y2 = 2k for some integers k  0. Answer questions 13 to 15 by using the given relation R. Question 13 Which one of the following is an ordered p…

Let R be the relation on Z (the set of integers) defined by (x, y)  R iff x 2 + y2 = 2k for some integers k  0. Question 15 R is not antisymmetric. Which of the following ordered pairs can be used together in a counterexam…

Qu 01: Model the following situations as (possibly weighted, possibly directed) graphs. Draw each graph, and give the corresponding adjacency matrices. (i) Rokey and Jane are friends. Rokey is also friends with Kevin and Ferdy. Jane …

Prove that (ab)n=anbn(ab)n=anbn is true for every natural number n

Use induction to prove for all n∈Nn∈N that n∑k=02k=2n+1−1.

Prove that 7n−17n−1 is a multiple of 6 for all n∈N.

1.Find a possible statistical problem inside your home relating to a discrete random variable lesson, explain in a paragraph form…..(2-3 sentence will do) 2.Find the possible random variable 3.Create a probability distribution 4.Co…

Exhibit a cycle of length 9 in this graph

Calculate the number of vertices in a full 5-ary tree with 45 internal vertices.Also find out the number of leaves.

Simplify the Boolean Function using K Map (5 pts each) and draw the logic diagram (5 pts each) a. F(a, b, c) = ∑ (0, 1, 2, 4, 5, 7) b. F(a, b, c, d) = ∑ (0, 1, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14) c. F (w, x, y, z) = ∏ (1, 2, 7, 11, 1…

In a school 100 students have access to three software packages A, B, C. Where 28 didn’t use any software, 8 used only package C, 26 used only package A,7 used package B, 10 used all three packages, 13 used both A and C a) Draw a …

Question 1: (2 marks) (C1) Let’s consider a propositional language where A =“I study data structure”, B =“I study Bioinformatics”, C =“I study programming”, D =“I study discrete math” a. “I study Bioinformatics if I study progra…

1. What are the converse, inverse and contrapositive of the statement “If you try hard, then you will win” Converse Inverse Contrapositive b. Determine whether this proposition is a tautology, contradiction or contingency using t…

(Direct proof) A claim is given as a quantified statement: “The product of two odd numbers is an odd number” a) (1 point) Write the domain of the variables: b) (4 points) Write the statement using quantifiers and an implication of pro…

Find the simplest form for the following boolean expressions: 1) (A.B'.C')+(A'.B'.C')+(A'.B.C')+(A'.B'.C) 2) (A'.B.C)+(A'.B.C)+(A.B.C')+(A.B'.C')+(A'.B.C')+(A'.B'.C') 3) (A+B+C)(A+B'+C')(A+B+C')(A+B+C') ( ' = Not )

£(x - 50) =60,£(y-20) = -20,£(x -50)^2 = 450,£(y - 20)^2 = 200,£(x - 50)(y - 20) = -100

Find the simplest form for the following boolean expressions: 1) (A.B'.C')+(A'.B'.C')+(A'.B.C')+(A'.B'.C) 2) (A'.B.C)+(A'.B.C)+(A.B.C')+(A.B'.C')+(A'.B.C')+(A'.B'.C') 3) (A+B+C)(A+B'+C')(A+B+C')(A+B+C') ( ' = Not )

£(x - 50) =60,£(y-20) = -20,£(x -50)^2 = 450,£(y - 20)^2 = 200,£(x - 50)(y - 20) = -100

What are the truth sets of the predicates P(x), Q(x), and R(x), where the domain is the set of integers and 1. P(x) is “|x| = 4,” 2. Q(x) is “x2 = 16,” 3. R(x) is “|x| = x”

State which of the following are not a function from R to R and why. (a) f(x) = 1/x (b) f(x) = 1/(1+x) (c) f(x) = (x)½ (d) f(x) = ±(x2+1)½ (e) f(x) = sin(x) (f) f(x) = ex

Find out which of the following functions from R to R are (i) One-to-one, (ii) Onto, (iii) One-to-one correspondence. (a) f: R—>R defined by f(x) = x (b) f: R—>R defined by f(x) = |x| (c) f: R—>R defined by f(x) = x + 1 (d) f…

Draw graphs of the following functions. (a) f: R—>R defined by f(x) = x (b) f: R—>R defined by f(x) = |x| (c) f: R—>R defined by f(x) = x + 1 (d) f: R—>R defined by f(x) = – 3x+4 (e) f: R—>R defined by f(x) = Floor(x) (f) …

What will be the inverse of the following functions from R to R? (a) f: R—>R defined by f(x) = x (b) f: R—>R defined by f(x) = x + 1 (c) f: R—>R defined by f(x) = – 3x+4 (d) f: R—>R defined by f(x) = x^3 (e) f: R—>R defined …

What will be the composition of two functions f ° g from R to R, (a) f(x) = 2x + 3, g(x) = 3x + 2 (b) f(x) = 2x + 3, g(x) = sin (x) (c) f(x) = sin (x), g(x) = 2x + 3 (d) f(x) = sin (x), g(x) = x2 (e) f(x) = |x|, g(x) = 2x + 3

Show that the following relations are Equivalence relations. Find out the Equivalence classes. (a) ρ = {(a, b) | a – b is an integer} on the set of real numbers R (b) R = {(a, b) | a = b or a = –b } on the set of integers Z (c) Con…

Show that the following relations are Partial order relations. (a) R on the set of integers Z, defined by, R = {(a, b) | a ≤ b} (b) R on the set of integers Z, defined by, R = {(a, b) | a ≥ b} (c) R on the set of positive integers …

Question 1. [20 Marks] a) Let p, q, and r be the propositions: p = "the flag is set" q = "I = 0" r = "subroutine S is completed" Translate each of the following propositions into symbols, using the l…

Every function is a relation, but the converse is not true.”--True or false? Justify with an example.

Show that a fuzzy relation R on a set U is anti-symmetric if and only if each of its alpha-cuts is an anti-symmettric relation on U

a) Write each statement in symbolic form using p, q and r. If I study, then I will not fail mathematics. If I do not play basketball, then I will study. But I failed mathematics. Then, test the validity of the fol…

Write down the relation R on the Power-set of S = {1, 2, 3, 4}, defined by, R = {(A, B) | A and B are subsets of S and they have the same cardinality} using any one of the methods: (i) set of ordered pairs, (ii) directed graphs, (ii…

(I) A relation R on a set S is called irreflexive, if no element in S is related to itself, that is if for every a in S, (a, a) is not in R. Which of the relations in Q. No. 5 is irreflexive?

A relation R on a set S is called asymmetric if (a, b) is in R implies that (a, b) is not in R. Which of the relations in Q. No. 5 is asymmetric?

Let R = {(0, 0), (1, 1), (1, 2), (1, 3), (2, 0), (2, 2), (3, 0)} be a relation on the set {0, 1, 2, 3}. Find the (a) Reflexive closure of R (b) Symmetric closure of R (c) Transitive closure of R

Show that the following relations are Equivalence relations. Find out the Equivalence classes. (a) ρ = {(a, b) | a – b is an integer} on the set of real numbers R (b) R = {(a, b) | a = b or a = –b } on the set of integers Z (c) Con…

Show that the following relations are Partial order relations. (a) R on the set of integers Z, defined by, R = {(a, b) | a ≤ b} (b) R on the set of integers Z, defined by, R = {(a, b) | a ≥ b} (c) R on the set of positive integers …

Question 1. [20 Marks] a) Let p, q, and r be the propositions: p = "the flag is set" q = "I = 0" r = "subroutine S is completed" Translate each of the following propositions into symbols, using the l…

Every function is a relation, but the converse is not true.”--True or false? Justify with an example.

Show that a fuzzy relation R on a set U is anti-symmetric if and only if each of its alpha-cuts is an anti-symmettric relation on U

a) Write each statement in symbolic form using p, q and r. If I study, then I will not fail mathematics. If I do not play basketball, then I will study. But I failed mathematics. Then, test the validity of the fol…

A fair six-sided dice is thrown and the scores are noted. Event X: The total of the two scores is 4. Even Y: The first score is 2 or 5. a) Construct the table by showing the sample spaces.

a) Freddie has 6 toys cars and 3 toy buses, all different. i) Freddie arranges these 9 toys in a line. Find the number of possible arrangements if the buses are all next to each other.

a) Freddie has 6 toys cars and 3 toy buses, all different. i) Freddie arranges these 9 toys in a line. Find the number of possible arrangements · if there is a car at each end of the line and no bus…

What is 100010002 - 00011011002

Suppose today is thursday 23 April 2020 use modular arithmetic to determine with a correct reason which day of the week will it be on 23 April 2036

Suppose U = {1, 2, 3, 4, 5, a, b, c} is a universal set with the subset A = {a, b, c, 1, 2, 3, 4}. Answer questions 1 and 2 by using the given sets U and A. Question 1 Which one of the following relations on A is NOT functional? …

Question 3 Let G and L be relations on A = {1, 2, 3, 4} with G = {(1, 2), (2, 3), (4, 3)} and L = {(2, 2), (1, 3), (3, 4)}. Which one of the following alternatives represents the relation L ￮ G = G; L? 1. {(2, 3), (3, 3)} 2. {(1,…

Answer questions 4 to 7 by using the given functions g and f. Hint: Drawing graphs of f and g before answering the questions, may assist you. Keep in mind that g  Z+  Q and f  Z+  Z+. Please note that graphs will not be asked …

Let A = {□, ◊, ☼, ⌂} and let # be a binary operation from A  A to A presented by the following table: # □ ◊ ☼ ⌂ □ □ ◊ ☼ ⌂ ◊ ◊ □ ◊ □ ☼ ☼ ◊ ☼ ⌂ ⌂ ⌂ □ ⌂ ⌂ Answer questions 10 and 11 by referring to the table for #. Question …

Question 14 Consider the truth table for the connective ‘’ with two simple declarative statements p and q. p q p  q T T T T F F F T F F F T (a) Which one of the given alternatives represents ‘’ as a binary operation on …

Question 15 Let p, q and r be simple declarative statements. Which alternative provides the truth values for the biconditional ‘’ of the compound statement provided in the given table? Hint: Determine the truth values of p → r, q…

Question 16 Consider the following quantified statement: ∀x ∈ Z [(x2 ≥ 0) ∨ (x2 + 2x – 8＞0)]. Which one of the alternatives provides a true statement regarding the given statement or its negation? 1. The negation ∃x ∈ Z [(x2 …

By constructing truth tables, decide which of the following are tautologies. (I) ~(P ^ ~P) (II) P implies ~P (III) (P ^ (p implies q)) implies q B) show that (P implies Q) implies R is logically equivalent to (~ P implies R) ^ (Q implies R)

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