Discrete Mathematics Answers

Let f be a function from A to B. Find f^-1

let P(x) denote the statement "x-3>5" . What are the truth values? P(2) P(8) P(9)

Generate the output in each set:

(a)    The set of all positive even numbers less than or equal to 10. (2 marks)

(b)    The set of all letters in the word "PURAKENCANA". (2 marks)

(c)    The set of all whole numbers greater than 3 and smaller than 16, and divisible by 3. (2 marks)

(d)    The set of all prime numbers divisible by 3. (2 marks)

(e)    The set of all numbers whose absolute value is equal to 7. (2 marks)

Let p and q be the propositions;

p: You drive over 110 kilometre per hour.

q: You get a speeding summons.

Write the following propositions using p and q and logical connectives.

a. You drive over 110 kilometre per hour, but you do not get a speeding summons.

(2 marks)

b. You get a speeding summon, but you do not drive over 110 kilometre per hour.

(2 marks)

c. If you do not drive over 110 kilometre per hour, then you will not get a speeding summon.

(2 marks)

d. Driving over 110 kilometre per hour is sufficient for getting a speeding summon.

(2 Marks)

a) Suppose that f is defined recursively by: and . Find f(1), f(2), f(3) and f(4)? [2 marks] b) For which positive integer n is it true that 2n >n³ !?

c) Prove your answer in

(b) above using mathematical induction.

d) Give a recursive definition of the sequence {a_n}, n = 1, 2, 3… if a_n = 2ⁿ + 1

e) Use your definition in (d) above to find a₁₀, and a₁₅

f) Let A = {1, 2, {{1,2}}}. Find the power set P(A)

1) Let  be a function from Z to R, such that , then  is

a) an increasing function   

b) a strictly increasing function

c) a decreasing function

d) an onto function

Consider the following assertions about the sets A, B and C. Write them down in the language of predicate logic. Use only the constructions of predicate logic (∀, ∃, ¬, ⇒, ∧, ∨) and the element of symbol (∈). Do not use derived notions (∩, ∪, =, etc.).

Hint “A is a subset of B” can be formalized as ∀x. x ∈ A =⇒ x ∈ B.

(i)

(ii)

(iii) The sets A and B are equal.

Every element of A is in the set B or the set C.

If A is disjoint from B then B and C overlap.

Let S={2, 4, 7}

and T={1, 3, 5}

. Find f(S×T)

if  

• f(x,y)=14x/3y
•  

In a high school, 50 students are surveyed and asked about the interest of the student. 23 students interested in English subject, 14 interested in Mathematics, and 11 in Urdu, 6 is in English an mathematics, 4 is in English and Urdu , and 5 is in English and Urdu. 15 students not interested in any subject. How many students interested in 3 subjects? Also represent your answer by using Venn diagram.

The study involved 55 students (23 male, 32 female) and used a mixed methods approach, involving a questionnaire with open ended questions, Likert scale questionnaire and interviews that aimed to determine students' perceptions of their performance. Five main areas were investigated with the open-ended questions: defining clinical reasoning; advantages and disadvantages of clinical case studies; the effectiveness of clinical case studies in comparison to real patients; and whether clinical case studies helped students working in a sports injury clinic. Students completed a 5-point Likert scale that asked three statements regarding the clinical environment. Following the questionnaire, a sample of 15 students were randomly selected for individual interviews. The results suggested that the students' responses were generally in favour of the use of clinical case studies to aid the development of confidence, communication and clinical reasoning.

1) Identify the premises of above argument