Lecture Notes - School of Mathematics

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0N1 (MATH19861) Mathematics for Foundation Year

Lecture Notes 23 Jan 2017

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0N1 • Mathematics • Course Arrangements • 23 Jan 2017

2

Contents Arrangements for the Course

I

7

Aims and description . . . . . . . . . . . . . .

7

Tests . . . . . . . . . . . . . . . . . . . . . . .

9

Examination . . . . . . . . . . . . . . . . . . .

11

Questions for students, email policy . . . . . .

12

Acknowledgements

13

Lecture Notes

14

1 Sets

14

1.1

Sets: Basic definitions . . . . . . . . . . . . .

14

1.2

Questions from students . . . . . . . . . . . .

17

2 Subsets; Finite and Infinite Sets

18

2.1

Subsets . . . . . . . . . . . . . . . . . . . . . .

18

2.2

Finite and infinite sets . . . . . . . . . . . . .

21

2.3

Questions from students . . . . . . . . . . . .

22

3 Operations on Sets

23

3.1

Intersection . . . . . . . . . . . . . . . . . . .

23

3.2

Union . . . . . . . . . . . . . . . . . . . . . .

23

3.3

Universal set and complement . . . . . . . . .

24

3.4

Relative complement . . . . . . . . . . . . . .

26

3.5

Symmetric difference . . . . . . . . . . . . . .

26

3.6

Boolean Algebra . . . . . . . . . . . . . . . .

26

3.7

Sample Test Questions . . . . . . . . . . . . .

28

3.8

Questions from Students . . . . . . . . . . . .

28

4 Set theory

33

0N1 • Mathematics • Course Arrangements • 23 Jan 2017

4.1

3

Proof of Laws of Boolean Algebra by Venn diagrams . . . . . . . . . . . . . . . . . . . . . .

33

4.2

Proving inclusions of sets . . . . . . . . . . . .

34

4.3

Proving equalities of sets . . . . . . . . . . . .

35

4.4

Proving equalities of sets by Boolean Algebra

36

4.5

Sample test questions . . . . . . . . . . . . . .

37

4.6

Additional Problems: Some problems solved with the help of Venn diagrams . . . . . . . . . . . . . . . . . . . . .

38

Questions from students . . . . . . . . . . . .

42

4.7

5 Propositional Logic

43

5.1

Statements . . . . . . . . . . . . . . . . . . . .

43

5.2

Conjunction . . . . . . . . . . . . . . . . . . .

43

5.3

Disjunction . . . . . . . . . . . . . . . . . . .

44

5.4

Negation . . . . . . . . . . . . . . . . . . . . .

45

5.5

Conditional . . . . . . . . . . . . . . . . . . .

45

5.6

Questions from students . . . . . . . . . . . .

47

6 Propositional Logic, Continued

50

6.1

Converse . . . . . . . . . . . . . . . . . . . . .

50

6.2

Biconditional . . . . . . . . . . . . . . . . . .

50

6.3

XOR . . . . . . . . . . . . . . . . . . . . . . .

51

6.4

Compound statements an