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1.1 sets, statements, and reasoning

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1.1 sets, statements, and reasoning

1. 1. MEMORARE Remember, O most gracious Virgin Mary, thatnever was it known that anyone who fled to thy protection, implored thy help, or sought thine intercession was left unaided. Inspired by this confidence, I fly unto thee, OVirgin of virgins, my mother; to thee do I come, before thee I stand, sinful and sorrowful. O Mother of the Word Incarnate, despise not mypetitions, but in thy mercy hear and answer me. AMEN
2. 2. 1.1Sets, Statements,and Reasoning
3. 3. SetA set is any collection of objects whichare known as elements of the set. A = { counting numbers} A is the set of counting numbers B = { 2, 4, 6, 8, } B is the set of elements 2, 4, 6, and 8
4. 4. A = { counting numbers}B = { 2, 4, 6, 8, }∴B is a subset of A becauseeach element in B is also in A.In symbol, B ⊆ A.
5. 5. U = { integers}A = { whole numbers}B = { negative numbers}∴B is a subset of Abecause each element in Bis also in A.
6. 6. In Geometry, geometricfigures such as lines andangles are actually setsof points.
7. 7. StatementA statement is a set of words andsymbols that collectively make a claimthat can be classified as TRUE or FALSE. 1. 10 > 17 2. The skin is the largest organ of the human body. 3. Stop singing. 4. 7 + 19 = 26
8. 8. Which sentences are statements?1 . Where do you live?2. 4 + 7 ≠ 53 . Quezon was the Philippine president.4 . X + 9 = 12 when X = 2
9. 9. VariableA variable is a letter representing anumber.Some statements contain one or morevariables.1. x + 5 = 62. a + b = 12
10. 10. Open statement or open sentenceIt is a statement that can be classifiedas true or false depending on thereplacement value of x.Find the replacement set.1. x + 5 = 62. 7 – 2x = –39
11. 11. Negation (∼ )The negation of a given statementP is a claim opposite the originalstatement.The negation of a true statement isfalse, and vice versa.If P is a statement, its negation is ∼P.
12. 12. Give the negation of eachstatement.1. 11 + 3 = 162. 18 > 43. ∠C is a right angle.4. I like vegetables.5. All jokes are funny.
13. 13. Give the negation of each statement.• Seoul is the Capital of South Korea.• Christopher Columbus crossed the Atlantic Ocean.• All jokes are funny.• No one likes me • Geometry is not my favorite subject.
14. 14. Compound StatementIt is a statement formed by combiningother statements ( P and Q) used abuilding blocks.Conjunction: P and QThe conjunction of P and QDisjunction: P or QThe disjunction of P and Q
15. 15. Compound StatementConjunction: P and QP: 3 + 8 = 11.Q: LSGH is located in San Juan City.P and Q3 + 8 = 11 and LSGH is located in SanJuan City.
16. 16. Compound Statements:1. 3 + 8 = 11 and LSGH is located inSan Juan City.2. 3 + 8 = 11 and LSGH is located inMandaluyong City.3. 3 + 8 = 38 and LSGH is located inMandaluyong City.4. 3 + 8 = 38 and LSGH is locatedinSan Juan City.
17. 17. Compound StatementDisjunction: P or QP: 3 + 8 = 11.Q: LSGH is located in San Juan City.P or Q3 + 8 = 11 or LSGH is located in SanJuan City.
18. 18. Compound Statements:1. 3 + 8 = 11 or LSGH is located inSan Juan City.2. 3 + 8 = 11 or LSGH is located inMandaluyong City.3. 3 + 8 = 38 or LSGH is located inMandaluyong City.4. 3 + 8 = 38 or LSGH is locatedinSan Juan City.
19. 19. The Table for Conjunction The Table for Disjunction--A conjunction is TRUE only when P and Q areBOTH TRUE.--A disjunction is FALSE only when P and Q areBOTH FALSE.
20. 20. Conditional Statement(or implication) “If P, then Q.”If an angle is a rightangle, then it measures90 . o
21. 21. If an angle is a rightangle, then it measures90 . oAll right angles measure90 . o (also a conditional statement)
22. 22. Conditional Statements: ( T or F )1. All right angles measure 90 . o2. An angle whose measure is 90ois a right angle.3. If angles are vertical, then theyare congruent.4. If I study hard, I will get highgrades.5. Love me, and I’ll love you back.
23. 23. Classify each statement as simple, conditional, a conjunction or a disjunction.1. If Alice plays, the volleyball team will win.2. The first-place trophy is beautiful.3. An integer is odd or it is even.4. I enrolled in Gold’s Gym and Fitness First.5. Charlie’s neat and Charlie’s sweet.6. You will be in trouble if you don’t change your ways.
24. 24. State the hypothesis and conclusion.1. If you go to the game, then you will have a great time.2. If two chords of a circle have equal lengths, then the arcs of the chords are congruent.3. Vertical angles are congruent when two lines intersect.4. Corresponding angles are congruent if two parallel lines are cut by a transversal.
25. 25. ReasoningReason is a term that refers to the capacity human beings haveto make sense of things, to establish and verify facts, and tochange or justify practices, institutions, and beliefs. It is closelyassociated with human activities as philosophy, science,language, mathematics, and art. The concept of reason issometimes referred to as rationality and sometimes asdiscursive reason, in opposition to intuitive reason.Reason or "reasoning" is associated with thinking, cognition,and intellect. Reason, like habit or intuition, is one of the waysby which thinking comes from one idea to a related idea. Forexample, it is the means by which rational beings understandthemselves to think about cause and effect, truth and falsehood,and what is good or bad.
26. 26. ReasoningIt is a process based on experience and principles thatallow one to arrive at a conclusion.1. INTUITION is an inspiration leading to the statementof a theory. It is a sudden insight allowing one to makea statement without applying any formal reasoning.2. INDUCTION an organized effort to validate a theory. (PARTICULAR TO GENERAL)3. DEDUCTION it is a formal argument that proves thetested theory. (GENERAL TO PARTICULAR)
27. 27. Use induction to state a conclusion.Several movies directed by LawrenceGarrison have won Awards, and manyothers have received nominations.His latest work, A Prisoner of Society,is to be released next week.Conclusion?
28. 28. While searching for a classroom, Tomstopped at an instructor’s office toask directions. On the officebookshelves are books titledIntermediate algebra, Calculus,Modern Geometry and LinearDifferential Equations. Conclusion?
29. 29. At a friend’s house, you seeseveral food items includingapples, pears, grapes, andbananas.Conclusion?
30. 30. On Monday, Matt says to you, “Andy hit hislittle sister at school today”. On Tuesday, Mattinforms you, “Andrew threw his Math bookinto the wastebasket during class.” OnWednesday, Matt tells you,”Because Andy wasthrowing peas in the school cafeteria, he wassent to the principal’s office.”Conclusion?
31. 31. Use deduction to state a conclusion, if possible.If the sum of two angles is 90 degrees,then these angles are“complementary.”Angle 1 measures 27 degrees andAngle 2 measures 63 degrees.Conclusion?
32. 32. Law of Detachment1. If P, then Q. premises2. PC. ∴ Q conclusion1. If a man lives in Davao, then helives in the Philippines.2. Dan lives in Davao.C. ∴Dan lives in the Phil.
33. 33. Law of Detachment1. If a man lives in Davao, then helives in the Philippines.2. Dan lives in the Philippines.C. ∴Dan lives Davao. Valid Argument Invalid Argument1. If P, then Q. 1. If P, then Q.2. P 2. QC. ∴ Q C. ∴ P
34. 34. Law of Detachment1.If an angle is a right angle, then itmeasures 90o.2. If you study hard, then you’ll gethigh grades.3. If an animal is a cat, then it makes ameow sound.
35. 35. Venn Diagram A Geometric figure representingsets of objects. It is created byJohn Venn , an Englishman. Each set is represented by aclosed figure such as circle orrectangle.
36. 36. Venn Diagram1. If a man lives in 1. If a man lives inDavao, then he lives Davao, then he lives inin the Philippines. the Philippines.2. Ken lives in Davao. 2. Ken lives in theC. ∴Ken lives in the Philippines.Phil. C. ∴Ken lives in Davao IS C IS P Q K? C etc D etc P K
37. 37. Venn Diagram1. If a man lives in Sucat, then he livesin Parañaque.2.If an angle is a right angle, then itmeasures 90o.3. If two angles are vertical, then theyare congruent.
38. 38. Venn DiagramConjunction (and)Intersection ( ∩ )-set of elements common to bothsetsDisjunction (or)Union (∪)-all the elements from both sets
39. 39. Venn DiagramU = { integers}A = { whole numbers}B = { negative numbers}C = { 2, 4, 6, 8, 10}Find:1. U ∩ A 4. U ∪ A 7. A ∩ B2. U ∩ B 5. U ∪ B 8. A ∩ C3. U ∩ C 6. U ∪ C 9. B ∪ C