Introduction to Proving

1. Introduction to Proving Statements Types of Reasoning Conditional Statements Writing – up Proofs

2. Statements • These are sentences that are either true or false but not both. Examples: p: The sun rises in the morning. q: 2.54 is an integer. r: 4 + 4 = 8 s: 3 is an even integer.

3. Not Statements… • Your place or mine? • Why is induction important? • Go to the Prefect’s Office. • Knock before entering! • It is hot today. • 3 + x = 5

4. Compound Statements • Sentences combining two or more ideas that can be assigned a truth value. (that is, true or false) Examples: • The number 5 is not an integer. • The number 4 is positive and the number 3 is negative. • If a set has n elements, then it has 2n subsets.

5. Compound Statements • 2n + n is a prime number for infinitely many n. • Every even integer greater than 2 is a sum of two prime numbers. • x + y = y + x, for all real numbers x and y. • It is not true that 3 is an even integer or 7 is a prime. • If the world is flat, then 2 + 2 = 4.

6. Negations ( ~ p ) Write the negation of the following statements: p: The sun rises in the morning. q: 2.54 is an integer. r: 4 + 4 = 8 s: 3 is an even integer

7. Negating Quantifiers

8. Examples • All students in this class will get 100 in Geometry. • Some calculators are solar powered. • No money grows on trees. • Some cows do not give milk.

9. Connectives Examples are and, or, if … then, if and only if etc. • Conjunction (Λ) – two simple statements joined by the word “and”. • Disjunction (V) – two simple statements joined by the word “or”.

10. Example: • Let p and q be simple statements. p: The number 3 is an even integer. q: The number 7 is prime. “ The number 3 is an even integer and the number 7 is prime.” “ The number 3 is an even integer or the number 7 is prime.”

11. Consider the ff. statements: • Albay is in Bicol and 5 + 5 = 10. • Albay is in Bicol and 5 + 5 = 11. • Albay is in Laguna and 5 + 5 = 10. • Albay is in Laguna and 5 + 5 = 12. Which of the following is true?

12. For Conjunction

13. Consider the ff. statements: • Marist is in Marikina or 5 + 5 = 10. • Marist is in Marikina or 5 + 5 = 11. • Marist is in QC or 5 + 5 = 10. • Marist is in QC or 5 + 5 = 12. Which of the following is true?

14. For Disjunction

15. Let p and q be statements given by: p: Triangle ABC has three sides. q: Triangle ABC has sides of the same length. Write the following in symbols. • Triangle ABC has 3 sides and its sides are of the same length. • Triangle ABC has 3 sides or its sides are of the same length. • Triangle ABC has 3 sides and its sides are not of the same length. • It is not true that triangle ABC has 3 sides and its sides are of the same length.

16. p: 6 is even ; q: 9 < 5 • p Λ q • p V q • p Λ ~ q • ~ p V q

17. Consider the statement: • If I will be elected, then I will be your voice in the administration. When will this person becomes true in his statement?

18. Conditional Statements • If – then statements • Formed by the connectives if and then • If – statements are called hypotheses • Then – statements are called conclusions

19. p –> q • If p then q • p implies q • p only if q • p is sufficient for q • q is necessary for p

20. Write each statement in “if-then” form. • All intelligent students can pass mathematics. • All right angles are equal. • Equal quantities multiplied by equal quantities are equal. • All triangles have three sides. • An obtuse angle is greater than an acute angle. • It is colder in winter.

21. Determine if the following statements are true. • If Boracay is in Bicol, then 3 + 3 = 6 • If Boracay is not in Bicol, the 3 + 3 = 6 • If Boracay is not in Bicol, then 3 + 3 = 4 • If Boracay is in Bicol the 3 + 3 = 4

22. Bi – Conditional Statements • “ p if and only if q” • p <-> q • p is necessary and sufficient for q. • p is equivalent to q. • It is true when p and q are both true or both false.

23. Check Your Understanding • If p<->q is true, what conclusion can be drawn from the statement: (~q v p) -> q ?

24. Converse of a statement • If the hypothesis and the conclusion in an implication are reversed, the new statement is called the converse of the given statement. • If q, then p. Example: Statement: Every dog is an animal. Converse: Every animal is a dog.

25. State the converse of each statement: • If two sides of a triangle are equal, then the angles opposite those sides are equal. • The diagonals of a rectangle are equal. • When it is raining, it is colder.

26. Inverse of a statement • The inverse of a statement , “If p, then q.” is “If not p then not q.” • ~ p -> ~ q Examples • If two sides of a triangle are equal, then the angles opposite those sides are equal. • Circles having equal radii are congruent. • When it is raining, it is colder.

27. Contrapositive of a statement • It is the converse of its inverse. • ~ q -> ~ p Examples • If two sides of a triangle are equal, then the angles opposite those sides are equal. • Circles having equal radii are congruent. • When it is raining, it is colder.

28. Consider the statement: The opposite angles of a parallelogram are equal. Give the following: • Conditional • Converse • Inverse • Contrapositive

29. Answers • Conditional • If two angles are opposite angles of a parallelogram, then they are equal. • Converse • If two angles of a parallelogram are equal, then they are opposite angles. • Inverse • If two angles of a parallelogram are not opposite angles, then they are unequal. • Contrapositive • If two angles of a parallelogram are unequal, then they are not opposite angles.

30. Types of Reasoning • Reasoning by Intuition • Reasoning by Analogy • Reasoning by Induction • Reasoning by Deduction

31. Reasoning by Analogy • It involves seeing similarities in different situations. If two situations are alike in certain ways, then perhaps they are also alike in others. Examples • Paeng, who was a good Mathematics student in grade school, concludes that he will also be a good Mathematics student in high school. • The peso is to the Philippines as the dollar is to the United States.

32. Reasoning by Induction • It involves making a general statement based on several instances in which this statement is true. Examples • Twelve is an even number and is divisible by 2. Six hundred eight is an even number and is divisible by 2. Therefore, all even numbers are divisible by 2.

33. Reasoning by Induction • It involves making a general statement based on several instances in which this statement is true. Examples 2. Tonichi is a third year student enrolled in Chemistry. Don is a third year student enrolled in Chemistry. Ross is also a third year student enrolled in Chemistry. Therefore, all third year students are enrolled in Chemistry.

34. Reasoning by Deduction • It involves reaching a conclusion from previously accepted statements. These statements contain the conditions which must be met for the conclusion to be true.

35. Reasoning by Deduction Examples • Our house is beside a large mango tree and at the back of a store. You have just passed a house beside a large mango tree but with no store at the back. You then conclude that it is not our house. • Any two right angles are congruent. Angle A and angle B are right angles. Therefore, angle A and angle B are conguent.