Discrete Mathematics Answers

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)

f: R—>R defined by f(x) = Ceiling(x)

(g)

f: R—>R defined by f(x) = x2

(h)

f: R—>R defined by f(x) = x3

Find out which of the following functions from R to R are (i) One-to-one, (ii) Onto, (iii) One-to-one corre￾spondence.

(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) = x2

(e)

f: R—>R defined by f(x) = x3

(f)

f: R—>R defined by f(x) = x – x2

(g)

f: R—>R defined by f(x) = Floor(x)

(h)

f: R—>R defined by f(x) = Ceiling(x)

(i)

f: R—>R defined by f(x) = – 3x+4

(j)

f: R—>R defined by f(x)= – 3x2 +7

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

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”

£(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 )

 (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 propositional functions: c) (15 points) Prove the statement by direct proof (Assume the hypothesis and derive the conclusion) 

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 programming and discrete math”

b. “I cannot study data structure when I do not study programming or discrete math ”

c. “I study data structure if and only if I do not study Bioinformatics”

Question 2: (2 marks)

Let’s consider a propositional language where

p means “Paola is happy”,

q means “Paola paints a picture”,

r means “Renzo is happy”.

Write English statement that corresponds to the following compound propositions:

1. p ∧ q → ¬ r

2. ¬ (p ∧ ¬q)

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 Venn diagram with all sets enumerated as for as possible.

b) If twice as many students used package A as Package C, write down a pair 

of simultaneous equations in x and y.

c) Solve these equation to find x and y.

d) How many students used package B.

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