# Mathematics

MAT1101 Assignment 2 Semester 1, 2014

Weight: 20% Total marks: 30

Due date: Friday 23May, 2014 23:55 AEST∗

Submission

• The assignment will be electronically submitted via StudyDesk.

• You are to submit your assignment as a Portable Document Format (PDF/A) file. Word files will not be accepted by the system. Instructions on how to save a Word 2010 document in PDF/A format are included on page 4.

• Hand-written and scanned assignments are perfectly acceptable, as long as they are submitted as a PDF file. You just need to ensure that the resulting scanned assignment is clearly legible.

• If you choose to typeset your assignment you must ensure that all math- ematical notation etc. follow standard mathematical conventions. The Learning Centre has some quick tip guides to typing Mathematics in Word (if you really need to typeset your assignment!).

• If you have trouble submitting your assignment etc., please contact the examiner (mat1101@www.sci.usq.edu.au.) or via phone ASAP.

Assignment instructions

• Show full working for each question. Give the marker every opportunity to see how you obtained your answers. Your mathematical reasoning is just as important as the final answer.

∗ Australian Eastern Standard Time

MAT1101 S1 2014

Question 1 [8 marks]

i) Construct the truth table for the following logical expression:

(p ∧ q) → r.

ii) Construct the truth table for the following logical expression:

¬p ∨ (q → r).

iii) Are the expressions in parts (i) and (ii) logically equivalent? Justify your answer.

iv) Using the laws of logic show that:

(p→ q) → (p ∧ r) ≡ p ∧ (q → r).

Question 2 [7 marks]

The sequence of numbers 2,10,18,26, . . . , (8n−6) is called an arithmetic series. The sum of the first n terms of this series is:

Sn = n 2

(8n − 4).

i) Show that this is correct for S2 (sum of the first two terms) and S3 (sum of the first three terms).

ii) Use the above result to calculate:

15∑ i=1

(8i − 6).

iii) Use the principle of mathematical induction to prove that:

2 + 10 + 18 + 26 + · · · + (8n − 6) = 4n2 − 2n.

Question 3 [12 marks]

i) Consider the following sets:

A = {1,3,5,7,9},B = {3,6,9} and C = {2,4,6,8}.

Determine the following sets constructed from the sets A,B, and C above.

a) A ∪ B

b) A ∩ B

c) A ∪ C

d) A ∩ C

e) A − B

f ) B ∪ C

g) B ∩ C

2 Due date: Friday 23May, 2014

S1 2014 MAT1101

ii) On a Venn diagrams similar to one shown in Figure 1, indicate the fol- lowing regions.

a) Q ∪ R b) P

c) P − (Q ∪ R) d) (P ∪Q) e) P ∩Q

ξ

P Q

R

Figure 1: Venn diagram illustrating the three overlapping sets (P,Q, and R) in relation to the Universal set (ξ).

Question 4 [3 marks]

Consider the two functions F : R→ R and F−1 : R→ R defined as follows:

F (x) = 3x + 2,∀x ∈ R,

and F−1(y) =

y − 2 3 ,∀y ∈ R.

Show that these functions are in fact the inverse of each other.

END OF ASSIGNMENT QUESTIONS

Due date: Friday 23May, 2014 3

MAT1101 S1 2014

Steps required to produce a PDF/A file from Microsoft Word 2010.

1. Save the document as a .docx file.

2. Go to the File menu and select Save As.

3. You should now see the dialogue box in Figure 2. In this dialogue make sure the Save as type is PDF, as shown in Figure 2.

Figure 2: Word 2010 Save As dialogue with the Save as type: PDF circled.

4. Select Options from the Save As dialogue box shown in Figure 2. A new window outlining extended options will appear as shown in Fig- ure 3. Make sure that ISO 19005-1 compliant (PDF/A) is selected as shown in Figure 3. Once completed click OK.

Figure 3: Word 2010 extend PDF options with the ISO 19005-1 compliant (PDF/A) check box highlighted.

5. Save the file with an appropriate ‘file name’. If it is your final assignment submission make sure you include your student number and course code in the file name.

4 Due date: Friday 23May, 2014