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- Minimize the following problems using the Karnaugh maps method. Z = f (A, B, C) = + B + AB + AC
- Using the Karnaugh map method obtain the minimal sum of the products and product of sums expressions for the function F (A, B, C, D) = Σ (1, 5, 6, 7, 11, 12, 13, 15).
- Let A = {1; 2; 3; 4; 5; 6; 7; 8; 9; 10} and B = {1; 2; 3; 4}. Let R be the relation on P (A) de fined by: For any X,Y element in P(A), XRY if and only if X-B = Y-B. How many equivalence classes are there? Explain
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- a) Let Fn denote the nth Fibonacci number. Show by induction that every natural number is expressible as a sum Fn1 +Fn2 +···+Fnk where ni −ni+1 > 1 for each i ≥ 1. b) Find the number of ways of tying up 7 different books into 4 bundle…
- a) Make a table of the values of the Boolean function f(x1,x2,x3) = x2 ⊕(x1 ∧x3) Write the function in DNF using the table. b) Find the general form of the solution to a linear homogeneous recurrence with constant coefficients for whi…
- a) Let A be an 8×8 Boolean matrix (i.e. every entry is 0 or 1). If the sum of the entries in A is 51, prove that there is a row i and a column j in A such that the entries in row i and in column j add up to more than 13. Further, show…
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- determine that wheter the functions from real numbers to real numbers are one to one f(n)=n^3 f(n)=n^2+1
- In a lottery, players win a large prize when they pick four digits that match, in the correct order, four digits selected by a random mechanical process. A smaller prize is won if only three digits are matched. What is the probability…
- If we reduce the number of elements by two, the number of permutations reduces thirty times. Find the number of elements.
- How many 8-bits sequences that start with the same two bits or their fourth and fifth bits are equal or end with the same two bits are there?
- Let D = {1, 2, 3}. The domain of the variables x and y will be D. Give an example of a predicate P(x, y) such that ∀x ∃y P(x, y) is true, but ∃y ∀x P(x, y) is false
- Let D = {1, 2, 3}. The domain of the variables x and y will be D. Is it possible to find a predicate P(x, y) such that ∃y ∀x P(x, y) is true but ∀x ∃y P(x, y) is false? Explain
- Determine if the following argument is valid using truth tables. p ∧ q → r __________ ∴ q → r
- Determine if the following argument is valid using truth tables. p −→ r r _______ ∴ p
- Determine if the following argument is valid using truth tables p ←→ q p∧ ∼ q _________ ∴ r
- Determine if the following argument is valid using truth tables p → q p → r __________ ∴ p → (q ∧ r)
- Determine if the following argument is valid using truth tables. p → (q ∨ r) ∼ q __________ ∴ p → r
- Let D = {1, 2, 3}. The domain of the variables x and y will be D. Give an example of a predicate P(x, y) such that ∀x ∃y P(x, y) is true, but ∃y ∀x P(x, y) is false
- Determine if the following argument is valid using truth tables. p ∧ q → r __________ ∴ q → r
- Let D = {1, 2, 3}. The domain of the variables x and y will be D. Is it possible to find a predicate P(x, y) such that ∃y ∀x P(x, y) is true but ∀x ∃y P(x, y) is false? Explain
- write the following boolean expressions in an equivalent sum of product canonical form in three variables x1, x2, and x3: 1. x1*x2 ? 3. (x1+X2)'*X3
- According to information obtained from mathematics department regarding three mathematics units done by 100 students, those who are doing calculus are 45, those doing discrete are 49 and those doing statistics are 38. Those doing calc…
- In how many ways can one select 7 member committee from 10 distinct persons if only three persons qualify to be chairperson?
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- Find a counter-example to the following statement: For all real numbers x > 1, 1/x^2+1 ≤ 1/2^x+1
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- Prove that for all integers a, b, c such that c =/= 0, if ac|bc then a|b.
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- Draw the graph represented by the adjacency matrix . (0 0 1 0 1 0 1 0 0)
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- Prove by mathematical induction the formula (1^3+2^3+3^3+4^3+....n^3)=(n^2(n+1)^2)/4
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- Given that p and q are propositions construct the truth table of p -> q p <-> q
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- Recall that a real number x is rational if x = p/q for integers p, q with q = ̸= 0. Prove that if x is rational then 1/(2x+1) is rational. Then prove that if 1/(2x+1) is rational then x is rational.
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- Let A = {1, 2, 3, 4} and let R be a relation on A such that R = {(1, 1),(2, 2),(3, 3),(4, 4),(1, 2),(2, 3),(1, 3)} Is R transitive? Symmetric? Reflexive?
- Let C = {1, 4, 5} and D = {2, 7}. • List the elements of C × D. • If there are 140 elements in D × A, how many elements are in A? • Give an example of a relation from D to C that is not a function. Draw a graph of the relation. Expla…
- Prove by mathematical induction that: Where "E" is the summation icon. n i E E j = 1/6n(n+1)(n+2) i = 1 j=1
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- Use mathematical induction to prove that 1^3 + 2^3 + ... + n^3 = =(n(n+1)/2)^2 for all integers n ≥ 1
- Let U be the set of positive integers 1, 2, 3, ... etc., A be the set of odd positive integers and B be the set of even positive integers. Verify De Morgan's laws.
- A school bus is transporting athletes home from an athletics meeting.of these athletes 15% took part in hurdles,20% took part in the long jump event ,35% took part in the relay races,30% ran 100- m sprint and 25% took part in high jum…
- Write the converse and the contrapositive of the following conditional: Conditional: If it is sunny then it is daylight. Converse: Contrapositive:
- A school bus is transporting athletes home from an athletics meeting. Of these athletes 15% took part in hurdles,20% took part in the long jump event 35%,took part in relay races,30% ran the 100-m sprint and 25% took part in the high-…
- The contrapositive of 'there exist y€z such that p(y) is true' is ' there exist x€z such that p(x) is true'. Is the statement true or false? Justify your answer.
- 1. Expand in a two-element universe (the elements are named 'a' and 'b') (a) ~(x) ((Fx v Gy) v Ka) (b) (x) ~ (Kx v Ka) (c) (Ex) (Cy v (Fx --> ~Ga)) 2. For the following wffs, indicate which variables are free and which are bound (…
- draw the graphs that have the following vertices and edges V = {a, b, c}, and E = {{a, b}, {b, c}}.
- draw a graph that has the following vertices and edges. V = {x1, x2, x3, x4} and E = {{x1, x3, }, {x1, x4}, {x2, x4}}
- How many 20 digit binary numbers have four 1's in them?
- Show that the following argument form is valid. p --> q q --> r ∴ p --> r
- a) A market researcher investigating consumers preference for three brands of beverages namely; coffee, tea and cocoa in Thika town gathered the following information. From a sample of 800 consumers, 230 took coffee, 245 took tea and …
- What is the value of k after the following code has been executed? k := 0 for i1 := 1 to n1 for i2 := 1 to n2 . . . for im := 1 to nm k := k + 1
- According to information obtained from mathematics department regarding three mathematics units done by 100 students, those who are doing calculus are 45, those doing discrete are 49 and those doing statistics are 38. Those doing calc…
- Give that the universal set • µ = (1,2,3,4,5,6,7,8,9,10) , p = (1,2,4,6,10) and Q = (2,3,6,9). Show that (P U Q)’= P’ n Q’
- “If it is a wild animal, then it is dangerous. If it is dangerous, then it will hurt you. However, it is not dangerous. Therefore, it is not a wild animal. Give a step-by-step argument using valid Rules of Inference
- Prove the following If a is odd and b is even, then a2 – b2 is an odd number
- P:the cow is old Q:it is dying Express each of these prepositions as an English sentence p v q. p ^q. p v(~q)
- If the domain is the set of words {violet, indigo, blue, green, yellow, orange, red}, which of the following sentences is/are true? (P)∀x(if x does not contain the letter ‘e’, then x contains the letter ‘n’) (Q) ∃x(if x contains the…
- 1. Let x = (1,2,3,4), Y = (2,3,5) and Z =(4,5,6). verify the following: a) x U y = y U x. b) (x U y) U z = x U (y U z).
- How many 10 digit binary numbers have four 1's in them?
- Go¨del′s completeness theorem asserts that --- The first order proof system with Peano's axioms proves every statement true in the standard model Peano's axioms form a consistent set of formulae The first order proof system can pro…
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- 7.The family of all the subsets of any set S is called a.the power set of S b.the null set of S c.the identity set of S d.the cardinality set of S 8.Let E = {2, 4, 6, ...}. What is the compliment of the set E? a.odd numbers b.prime n…
- 1. For integers a and b,if ab is odd, then a and b are odd. 2. If xy=(x +y)^2 / 4 ,then x=y. prove the following 2 statements, state the method used and explain all necessary steps. 3.Suppose that factorial is the Python function de…
- Prove or disprove: If A, B, and C are nonempty sets, and A×B = A×C, then B = C.
- If the domain of discourse is all integers, find a counterexample*, if possible, to the following universally quantified statements: a. ∀x∃y(x = 1/y) b. ∀x∃y(y2 −x < 100) c. ∀x∀y(x2= y3)
- Suppose g : A → B and f : B → C are functions. a. Show that if f ◦g is onto, then f must also be onto. b. Show that if f ◦g is one-to-one, then g must also be one-to-one. c. Show that if f ◦g is a bijection, then g is onto if and only…
- Write in expression in p, q, and logical connectives which gives the following truth table: p q ? p= T T T F, q= F T F F, s= F T F F
- Express the negation of the following statements WITHOUT using the negation symbol: a. ∀x(−2 < x <3) b. ∀x(0 ≤ x <5) c. ∃x(−4 ≤ x ≤1) d. ∃x(−5 < x <−1) e. ∀x∃y(x2 < y)
- Use the principle of induction Prove that 2 1 n 2 n 2 n n 1 > + ∀ > − ,
- Show that 1n^3+2n+3n^2 is divided by 2 and 3 for all positive integers n
- which of the following statement is true a. \\((p\\wedge q)\\vee (p\\vee r)=p\\vee (q\\wedge r)\\) b. \\((p\\vee q)\\wedge (p\&b…
- What reads "the goods are are standard if and only if the goods are expensive"?
- Let A = {2, 3, 4} and B = {6, 8, 10} and define a relation R from A to B as follows: For all (x, y)∈ A ×B, (x, y)∈ R means that is an integer. a. Is 4 R 6? Is 4 R 8? Is (3, 8) ∈R? Is (2, 10) ∈R?
- At the Keep in Shape Club, 35 people swim, 24 play tennis, and 27 jog. Of these people, 12 swim and play tennis, 19 play tennis and jog, and 13 jog and swim. Nine people do all three activities. How many members are there altogether?
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