Solved: 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

Describe the Hasse diagram formed by the Relation "x is a divisor of y" for the set A = {1, 3, 6, 12, 24, 48}

Determine whether the given relation is reflexive, symmetric, transitive, or none of these. R is the ”greater than or equal to” relation on the set of real numbers: For all x, y ∈ R , xRy ⇐⇒ x ≥ y

determine how many bit strings of length can be formed, where three consecutive 0s are not allowed

Use a Venn diagram to illustrate the set of all months of the year whose names do not contain the letter R in the set of all months of the year.

Let p and q be the propositions defined as below. p : It is below freezing. q : It is snowing.

Show that a complete graph with n vertices has n(n -1) 2 edges

Show that for any real number x, if x2 is odd, then x is odd.

Let p, q and r be statements. Use the Laws of Logical Equivalence and the equivalence of → to a disjunction to show that: ∼ ((p ∨ (q →∼ r)) ∧ (r → (p∨ ∼ q))) ≡ (∼ p ∧ q) ∧ r.

Let p, q and r be statements. Suppose you know that the statement form ((q → p) ∨ r) ∨ (∼ r ∧ p) is false. What can you conclude about the truth values of the three statement variables?

Model two contextualized problems using binary trees both quantitatively and qualitatively

Prove the following sentences using any prove method [ the most appropriate one]: a) If r is rational and s is irrational, prove that 2r+s is irrational. b) If t and s are integers and t × s is even, then t is even or s is even.

Prove that (a ∧ (b → ¬a)) → ¬b is a tautology.

Prove that the following argument is valid. Write all the necessary steps a)Rashid didn’t perform well in the subject, but he was present in every class. b)Every student who completed all the assignments, performed well in the subje…

Determine whether each of the following compound propositions is satisfiable. Justify your answer. 1. (a) (pv-g)^(-pVg)^(-p^ g) (b) ()^(-)^(-p)^(---) (c) (pvqv-r) ^ (pv-gv-8) ^ (pv-rv-8)^(-pv-gv-) A (pvqV-s)

2) Construct a circuit using NOT gates, OR gates and AND gates of the following proposition (P and -r)or(-q and r)

3) Using basic propositional equivalence (which we did in the class) show that (pr) ^ (4) and (pvg) → are logically equivalent. (Don't use truth table for this problem) O U

2.1 Differentiate f(x) = (2x3 + 3x2 )(x2 + 5x3 +5) using product rule. (5) 2.2 Differentiate f(x) = (8x 3 + 4x )(2x 2 +5) using product rule. (5) 2.3 Differentiate f(x) = 5𝑥³+7𝑥 2𝑥²−4𝑥+5 using quotient rule. (5) 2.4 Diffe…

2fn-f(n-2) = fn+1 for n>3

The trivial negation of a proposition is: “It is not the case that [proposition]." Write two negations of the following, one trivial and one not trivial. (a) It is snowing. (b) At least 3 inches of snow fell yesterday. (c) 1…

Let P (x,y) denote the sentence 2x+y=5. what are truth value of the following domain of x and y is the set of all integers?

Sketch the Venn diagrams for each of these combinations of the sets A,B,C. (A ∪ B) ∩ C

a) A study was conducted on some 400 employees of a certain company to establish the favorite beverage from among Coffee, Tea and Cocoa. The following was observed: 145 employees preferred Coffee, 165 preferred Tea while 185 prefe…

A Survey of 118 persons was conducted at TCC, and it was found at 24 persons carried a cell phone, 59 persons carried a tablet computer and 12 carried both a cell phone and a tablet. how many people carried a cell phone or a tablet? …

What do universal set, S, have 91 elements. A and B are subsets of S. Set a contains 20 elements and CB contains 44 elements. If sets A and B have 6 elements in common, how many elements are in A but not in B?

Let A = {a,b,d,e,g,f,1,2,3}, B = {1,2,3}, and C = {1,2,3,a,d,g}. Find the following: (i) A ՈB Ս C (ii) A-B (iii) (B Ո C) x (A Ո C) (iv) P(B Ո C) (v) |CxA|

(a) Let A = {0,1, 2, 3, 4, 5}, B = {1, 2, 3, 4, 5, 6}, and consider the relation R = {(a, b) э A x B | a2+b2 < 30}. (i) List all the elements of the relation R and give its cardinality |R|. (3 marks) (ii) Find the domain and range …

(b) Let A = {5, 6, 7, 8}, B = { 4, 6, 7} and the relations R1 = {(a, b) | a эA, bэB and a > b} R2 = {(a, b) | a э A, b э B and (a – b)2 <=6} (i) Find the sets of ordered pairs in R1, R2 and give their cardinalities |R1|, |R2|. (2 m…

5a) Explain whether each of the following relations on the set of real numbers is a function or not. For those (if any) that are indeed functions say whether they are one-to-one and/or onto. (2 marks) i) y = f(x) = 2x2+1 xэR, y эR i…

Question 6 (6 marks) 6a) Draw the Venn diagrams for each of these combinations of the sets A, B, and C: (i) (A – C)C ⋂ ( B- C)C (3 marks) (ii) (A – C) U (C – B)(3 marks) End

Find a formula for 1+1+⋯+ 1 1 ⋅ 2 2 ⋅ 3 n(n + 1) by examining the values of this expression for small values of n. b) Prove the formula you conjectured in part

Prove that 12 +32 +52 +⋯+(2n+1)2 = (n+1)(2n+1)(2n + 3)∕3 whenever n is a non negative integer.

1. Is the function 𝑓: ℤ → ℤ 𝑓(𝑥) = 𝑥 2 + 3 injective, surjective or bijective? Prove your assertions

1. Show if satisfy the algebraic axioms: identity and inverse under addition. 2. Show if satisfy all (communicative, associative, identity, and inverse) the algebraic axioms under both multiplication and addition. 3.…

3a²n−1

find 200 sum k=100^k

Check whether the compound proposition (p ∨￢q) ∧(q ∨￢r) ∧(r ∨￢p) ∧(p ∨q ∨r) ∧(￢p ∨￢q ∨￢r) is satisfiable or not?

Given a function f from {1, 2,...,n}to the set of integers, determine whether f is one-to-one.

Why is f not a function from R to R if a) f (x) = 1/x? b) f (x) =√x? c) f (x) = ±√(x^2+1)?

BSIT section 1T is composed of 11 male students and 19 female students, how many choices do you have if there will be one (female) muse and one (male) escort

Mary’s Online Shop sells clothes for plus size women. It has 4 kinds of formal blazers, 7 kinds of inner blouses, and 9 kinds of designer jeans. Find the number of ways a person can buy an item

Determine the truth value of each of the following statements:/5mrks (i) Paris is in France or 4+4=8 (ii) 4+4=9 and5+8=11 (iii) m+2is the unit of area or Kg is the unit of mass (iv) If Kigali is capital city of Rwanda and Kampal…

1 + 1 =

¬𝑝 ∧ (𝑝 ∨ 𝑞)] ⇒ 𝑞

Given a set of five distinct science books, three distinct math books and two distinct history books. a. In how many ways can these books be arranged on a shelf?

Establish the validity of the argument with the premises p -> (q -> r) , p \/ s , t ->q , ~s and ~r -> ~t

Let Q(x) be the statement “x + 1 > 2x.” If the domain consists of all integers, what are these truth values? a) Q(0) b) Q(−1) c) Q(1) d) ∃xQ(x) e) ∀xQ(x) f) ∃x¬Q(x) g) ∀x¬Q(x)

A 1 team1 of :ll.1 players is so be chosen from 15 1 members. 11 n how ways rcan this, be done If I.. One pairticular player is always incliuded.

∃𝑥 (𝑥 + 1 = 0 ∨ 𝑥 + 2 = 0) write the negation of the following

whether each of these functions is a bijection from R to R. f (x) = x 3 -1

A) The probability that a free-lancer gets an electric contract is 2/7 and that of his getting a plumbing contract is 3/5. If the probability of his getting at least one of the contracts is 5/7. What is the probability of his gettin…

Show that ¬ (P\iff⟺Q)\iff⟺(P V Q) Λ ¬(P Λ Q) \iff⟺(P Λ ¬Q) V (¬ P Λ Q) without using truth table

a recursive function is defined by can=2an-1 with a0=1. find the value of a3

Find these terms of the sequence (An}, where An = 2(3)n + 5

Find the transitive closure A={1,2,3,4,5} R={ (2,3),(3,5),((1,2),(2,5),(1,3),(4,5),(5,2)} by using Warshall’s Algorithm.

Determine whether each of these functions is a bijection from R to R?f (x) = (x2 + 1)/(x2 + 2)

Draw the Hasse diagram for the “divides” relation on {2, 3, 5, 10, 11, 15, 25, 36, 42, 108}

Define a bijective function. Explain with reasons whether the following functions are bijective or not. Find also the inverse of each of the functions. i. f(x) = 4x+2, A=set of real numbers ii. f(x) = 3+ 1/x, A=set of non zero real nu…

Let f and g be functions from the positive real numbers to positive real numbers defined by f(x) = [2x] g(x) = x2. Calculate f o g and g o f.

If R is a relation defined on the set Z by a R b if a-b is a non negative even integer. Determine if R define a partial order and total order.

13. a) Consider f; Z+ → Z+ define by f(a) =a2. Check if f is one-to-one and / or into using suitable explanation. b) What is a partial order relation? Let S = { x,y,z} and consider the power set P(S) with relation R given by set incl…

Let A,B,C єR2 where A = { (x,y) / y = 2x + 1} , B = { (x,y) / y = 3x} and C = { (x ,y) / x - y = 7} . Determine each of the following: i. A B ii. B intersection C complement 3.

Determine whether the following relations are injective and/or subjective function. Find universe of the functions if they exist. i. A= v,w,x,y,z, B=1,2,3,4,5 R= (v,z),(w,1), (x,3),(y,5) ii. A = 1,2,3,4,5 B=1,2,3,4,5 R = (1,2),(2,3),(…

Let X = {1,2,3,4,5,6,7} and R = {x,y/x–y is divisible by 3} in x. Show that R is an equivalence relation. b) Let A = {1,2,3,4} and let R = {(1,1), (1,2),(2,1),(2,2),(3,4),(4,3), (3,3), (4,4)} be an equivalence relation on R. Determine…

Let X = {a, b, c} defined by f : X X such that f = {(a, b), (b, a), (c,c)}. Find the values of f–1, f2 and f4. b) Let L = {3, 4, 12, 24, 48, 82} and the relation < be defined on L such that x < y if x divides y. Draw the Hasse diagra…

Boolean expressions to minimal number of literals: 1. x'y' + xy + x'y 2. (x + y) (x + y') 3. x'y + xy' + xy + x'y' 4. x' + xy +xz' + xy'z' 5. A'C' + ABC + AC' 6. (x'y' + z)' + z + xy + wz 7. A'B (D' + C'D) + B (A + A'CD)

Find the complement of the following expression using dual of a function: 1. xy' + x'y 2. (AB' + C) D' + E 3. AB (C'D + CD') + A'B' (C' +D) (C + D') 4. (x + y+ z) (x' + z') (x + y)

Create the equivalent logic circuit of the following logic expression: 1. Q = (A + B) . (C +D)' 2. F1 = (A + BC') . D 3. Q = [(A + B)' . C] +B . C

Determine whether each of these functions is a bijection from R to R. f (x) = (x2 + 1)/(x2 + 2)

Q#: A binary operation ∇ on [0,1] is a t-norm if and only if x∇(y∇z)=(x∇y)∇z

Use a proof by contradiction to show that there is no rational number r for which r+r+1=0. (Assume that r=a/h is a root, where a and b are integers and a/b is in lowest terms.obtain an equation involving integers by multiplying by th…

Let X = {a, b, c} defined by f : X ®X such that f = {(a, b), (b, a), (c,c)}. Find the values of f–1, f2 and f4

Consider f; Z+ → Z+ define by f(a) =a2. Check if f is one-to-one and / or into using suitable explanation

Show that if eight people are in a room, at least two of them have birthdays that occur on the same day of the week.

Give a relation which is both a partially ordered relation and an equivalence relation on a set.

Let X = {a, b, c} defined by f : X X such that f = {(a, b), (b, a), (c,c)}. Find the values of f–1, f2 and f4. b) Let L = {3, 4, 12, 24, 48, 82} and the relation < be defined on L such that x < y if x divides y. Draw the Hasse diag…

Determine whether each of these functions is a bijection from R to R. f (x) = (x2 + 1)/(x2 + 2)

Determine all eigenvalues and the corresponding eigenspaces for the matrix 𝐴 = [ −9 4 4 −8 3 4 −16 8 7 ]

Employ the Gauss-Seidel method, solve the system. 10𝑥 + 𝑦 + 𝑧 = 12 2𝑥 + 2𝑦 + 10𝑧 = 14 2𝑥 + 10𝑦 + z=13

Let p, q and r be the propositions p : Grizzly bears have been seen in the area. q : Hiking is safe on the trail. r : Berries are ripe along the trail. Write these propositions using p, q, and r and logical connectives (including …

Let set A = {1, 2, 3, 4} and B = {5, 6, 7, 8} whereby relation R1 = {(a, b) | a = b - 1 } and R2 = {(a, b) | a + b ≥ 3}. a is an element of set A and b is an element of set B. Write the relation for R1 and R2 where these …

Let A,B,C єR2 where A = { (x,y) / y = 2x + 1} , B = { (x,y) / y = 3x} and C = { (x ,y) / x - y = 7} . Determine each of the following: i. A intersection ii.( B intersection C ) complement iii. A complent union C complent iv. B comple…

If a function is defined as f(x,n) mod n. Determine the i. Domain of f ii. Range of f iii. G(g(g(g(7)))) if g (n) = f(209, n).

Let X = {1,2,3,4,5,6,7} and R = {x,y/x–y is divisible by 3} in x. Show that R is an equivalence relation.

Let A = {1,2,3,4} and let R = {(1,1), (1,2),(2,1),(2,2),(3,4),(4,3), (3,3), (4,4)} be an equivalence relation on R. Determine A/R.

Let X = {a, b, c} defined by f : X

Let L = {3, 4, 12, 24, 48, 82} and the relation < be defined on L such that x < y if x divides y. Draw the Hasse diagram.

Show that the functions f: R -->1 , infinity -->R , defined by : x=3^2x +1 , x= log x-1 are inverse of one another.

Define a bijective function. Explain with reasons whether the following functions are bijective or not. Find also the inverse f(x) = (2x+3) mod7, A=N7

Check the extension of d given by d(x,y) = d(x,a) + 1 + d(b,y) where x belongs to X and y belongs to Y/ X

Determine whether the following relations are injective and/or subjective function. Find universe of the functions if they exist. i. A= v,w,x,y,z, B=1,2,3,4,5 R= (v,z),(w,1), (x,3),(y,5) ii. A = 1,2,3,4,5 B=1,2,3,4,5 R = (1,2),(2,…

If a function is defined as f(x,n) mod n. Determine the i. Domain of f ii. Range of f iii. G(g(g(g(7)))) if g (n) = f(209, n).

Let X = {a, b, c} defined by f : X

Show that the functions, defined by f:R->(1,infinity) ->R f(x)=3^2x+1, g(x)=log(x-1) are inverse of one another.

Let A={2, {2}, (2)2}, B= {2, {2{, {{2}} and C={2}. Evaluate the truth and falsity of the following statements.

write the following sets by listing their elements between braces

transitive closure of{(1,3),(5,5),(1,6),(3,3),(5,6)}

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

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Question ID: mtid-5-stid-8-sqid-2874-qpid-1498