The document discusses key concepts in discrete mathematics and logic. It defines propositions as basic building blocks represented by letters, and connectives as operators used to combine propositions like conjunction, disjunction, negation, and conditional statements. It provides truth tables showing the truth values of propositions under different connectives and examples of applying these logic rules.

1. DISCRETE MATHEMATICS

2. LOGIC

3. All Mathematicians wear sandals Anyone who wears sandals is an algebraist Therefore, all mathematicians are algebraist

4. LOGIC Is the study of reasoning Specifically concerned with whether reasoning is correct. Focuses on the relationship among statements as opposed to the content of any particular statement.

5. Propositions Typically expressed as a declarative sentence Basic building blocks of any theory of logic Represented by lowercase letters such as p, q and r.

6. Connectives Used to combine propositions

7. Kinds Of Connectives CONJUCTION – denoted by (read as “p and q”) DISJUNCTION – denoted by ( read as “p or q” ) NEGATION -- denoted by (read as “not p”)

8. Kinds Of Connectives CONDITIONAL STATEMENT denoted by p  q (read as If p, then q.)

9. Truth Table Of A Proposition Made up of individual proposition ... , lists all possible combinations of truth values for .... .T denotes true and F denotes false for such combination lists of the truth value of p .

10. CONJUCTION p q T T T T F F F T F F F F

11. DISJUNCTION p q T T T T F T F T T F F F

12. NEGATION p -p q -q T F T F

13. IF-THEN STATEMENTS The most commonly used connectives. It also known as conditional statements or implications.

14. IF-THEN STATEMENTS It consist of the following: Premise – the “if” part Conclusion – the “then” part Represented by the following: If p, then q p -> q Where p and q are the premise and conclusion respectively.

15. IF-THEN STATEMENT Example: If one angle of a triangle is a right triangle , then the other two angles of the triangle are acute angles. premise conclusion

16. IF-THEN STATEMENT Example: If one angle of a triangle is a right angle , then the other two angles of the triangle are acute angles. p q

17. IF-THEN STATEMENTS it can only be false when the premise is true but the conclusion is false.

18. If

19. If If a picture paints a thousand words Then why can't I paint you? The words will never show For you I've come to know. If a face could launch a thousand ships Then where am I to go? There's no one home but you And now you've left me too.

20. And when my love for life is running dry You come and pour yourself on me If a man could be two places at one time I'd be with you. Tomorrow and today Beside you all the way If the world should stop revolving Spinning slowly down to die.

21. I'd spend the end with you And when the world was through... Then one by one, the stars would all go out. Then you and I, would simply fly away.

22. CONDITIONAL STATEMENT p q p  q T T T T F F F T T F F T

23. BICONDITIONAL STATEMENT It is denoted by : read as “p if and only if q”

24. BICONDITIONAL STATEMENT p q T T T T F F F T F F F T

25. Example: p: Today is Monday. q: it is raining. CONJUNCTION DISJUNCTION NEGATION CONDITIONAL STATEMENT BICONDITIONAL STATEMENT

26. CONJUNCTION p: Today is Monday. q: it is raining. Today is Monday AND it is raining.

27. DISJUNCTION p: Today is Monday. q: it is raining. Today is Monday OR it is raining.

28. NEGATION p: Today is Monday. q: it is raining. -p: Today is NOT Monday. -q: It is NOT raining.

29. CONDITIONAL STATEMENT p: Today is Monday. q: it is raining. p  q IF today is Monday, THEN it is raining.

30. BI CONDITIONAL STATEMENT p: Today is Monday. q: it is raining. Today is Monday IF AND ONLY IF it is raining.