Lecture Notes - School of Mathematics

Jan 4, 2018 - If A and B are empty sets then A = B. Proof. ∗. The sets A and B are equal because they cannot. * The word proof indicates that an ar- gument establishing a theorem or other statement will follow. be non-equal. Indeed, for A and B not to be equal we need an element in one of them, say a ∈ A, that does not ...
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0N1 (MATH19861) Mathematics for Foundation Year

Lecture Notes 04 Jan 2018

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0N1 • Mathematics • Course Arrangements • 04 Jan 2018

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Contents Arrangements for the Course

I

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

8

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

10

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

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Questions for students, email policy . . . . .

13

Acknowledgements

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Lecture Notes

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1 Sets

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

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

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2 Subsets; Finite and Infinite Sets

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2.1 Subsets . . . . . . . . . . . . . . . . . . . . .

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

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

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3 Operations on Sets

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3.1 Intersection . . . . . . . . . . . . . . . . . . .

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3.2 Union . . . . . . . . . . . . . . . . . . . . . .

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

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3.4 Relative complement . . . . . . . . . . . . . .

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3.5 Symmetric difference . . . . . . . . . . . . . .

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3.6 Boolean Algebra . . . . . . . . . . . . . . . .

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

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3.8 Questions from Students

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4 Set theory

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0N1 • Mathematics • Course Arrangements • 04 Jan 2018

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

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

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

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4.4 Proving equalities of sets by Boolean Algebra

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

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

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

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5 Propositional Logic

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5.1 Statements . . . . . . . . . . . . . . . . . . .

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5.2 Conjunction . . . . . . . . . . . . . . . . . . .

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5.3 Disjunction . . . . . . . . . . . . . . . . . . .

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5.4 Negation

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5.5 Conditional . . . . . . . . . . . . . . . . . . .

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

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6 Propositional Logic, Continued

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6.1 Converse . . . . . . . . . . . . . . . . . . . .

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6.2 Biconditional . . . . . . . . . . . . . . . . . .

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6.3 XOR . . . . . . . . . . . . . . . . . . . . . . .

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6.4 Compound statements and truth tables

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6.5 Tautologies . . . . . . . . . . . . . . . . . . .

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6.6 Contradictions . . . . . . . . . . . . . . . . .

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6.7 Matching brackets: a hard question . . . . . .

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