Discrete Mathematics Answers

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

ii) y = g(x) = 1/(x+1) (xэR, y эR , x != -1)

iii) Let h be a function from X = {1, 2, 3, 4} to Y = {a, b, c, d}.

h(1) = d, h(1) = c, h(2) = a, h(3) =b, and h(4) = b.

5b) Does either f or g have an inverse? If so, find this inverse. (1 marks)

5c) Find the composite functions f 。g and g。f . (2marks)

Question 6

(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 marks)

(ii) Draw the directed graphs of R1, R2. (2 marks)

(iii) Give the Boolean-matrix representations of R1, R2. (2 marks)

(c) Explain what are Reflexive, Symmetric relations using examples. Each relation should contain at least three elements. (2 marks)

(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 of the relation R. (2 marks)

(iii) Find the inverse relation R-1 (2 marks)

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|

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?

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.

1. how many people carried a cell phone or a tablet?
2. how many people carry neither a cell phone nor a tablet?
3. how many people carried a cell phone only?
4. how many people carried a tablet but not a cell phone?

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?

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 + 2 = 3.

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