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Logic

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Logic

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  1. Logic

  2. A conjecture is an educated guess that can be either true or false. A statement is a sentence that is either true or false but not both. Often represented by a letter such a p, q or r. The Truth value is the truth or falsity of a statement.

  3. Negations The negation of a statement says it has the opposite meaning. symbol ~p or ~q read “not p or not q” Example: p: Suffolk is a city in Virginia. The negation would be: ~p: Suffolk is not a city in Virginia.

  4. Compound Statement A compound statement is two or more statements that are joined together. Example: p: Richmond is a city in Virginia. q: Richmond is the capital of Virginia. p and q: Richmond is a city in Virginia and Richmond is the capital of Virginia.

  5. Conjunction A conjunction is a compound statement formed by joining two or more statements with the word “AND”. Symbolic representation: “read” p and q * A conjunction is true IFF both statements are true.*

  6. Example:use the following statements to write a compound statement for each conjunction then find it’s truth value. p: One foot is 14 inches. q: September has 30 days r: A plane is defined by 3 non-collinear points. a) p and q One foot is 14 inches and September has 30 days. b) A plane is defined by 3 non-collinear points and one foot is 14 inches. c) September does not have 30 days and a plane is defined by 3 non-collinear points. d) One foot does not have 14 inches and a plane is defined by 3 mom-collinear points.

  7. Disjunction A disjunction is a compound statement that joins two or more statements with the word “or”. Symbolic representation: “read” p or q *A disjunction is true if at least one of the statements are true.*

  8. Example:use the following statements to write a compound statement for each disjunction then find it’s truth value. p: is proper notation for “line AB” q: centimeters are metric units. r: 9 is prime number a) p or q is proper notation for “line AB” or centimeters are metric units. b) Centimeters are metric units or 9 is a prime number.

  9. Conditional Statement Definition: A conditional statement is a statement that can be written in if-then form. “If _____________, then ______________.” Lesson 2-1 Conditional Statements “if p, then q”. Symbolic Notation p → q

  10. Conditional Statement Conditional Statements have two parts: The hypothesis is the part of a conditional statement that follows “if”(Usually denoted p.) The hypothesis is the given information, or the condition. Lesson 2-1 Conditional Statements The conclusionis the part of an if-then statement that follows “then”(Usually denoted q.) The conclusion is the result of the given information.

  11. Example Write the statement “ An angle of 40° is acute.” Hypothesis– An angle of 40° Represented by : p Conclusion – is Acute Represented by : q If – Then Statement – If an angle is 40°, then the angle is acute.

  12. Example Identify the Hypothesis and Conclusion in the following statements: • If a polynomial has six sides, then it is a hexagon. H: A polygon has 6 sides C: it is a hexagon • Tamika will advance to the next level of play if she completes the maze in her computer game. H: Tamika Completes the maze in her computer game. C: She will advance to the next level of play. p q

  13. Forms of Conditional Statements Conditional Statements: Formed By: Given Hypothesis and Conclusion. Symbols: p → q Examples: If two angles have the same measure then they are congruent.

  14. Forms of Conditional Statements Converse: Formed By: Exchanging Hypothesis and conclusion of the conditional. Symbols: q → p Examples: If two angles are congruent then they have the same measure.

  15. Forms of Conditional Statements Inverse: Formed By: Negating both the Hypothesis and conclusion of the conditional. Symbols: ~p →~q Examples: If two angles do not have the same measure they are not congruent.

  16. Forms of Conditional Statements Contra - positive: Formed By: Negating both the Hypothesis and conclusion of the Converse statement. Symbols: ~q →~p Examples: If two angles are not congruent then they do not have the same measure.

  17. Logically Equivalent Statements - are statements with the same truth values. Example: Write the converse, inverse and contra - positive of the following statement: Conditional: If a shape is a square, then it is a rectangle. Converse: If a shape is a rectangle, then it is a square. Inverse: If a shape is not a square, then it is not a rectangle. Contra-positive: If a shape is not a rectangle, then it is not a square.

  18. Try This: Example: Write the converse, inverse and contra - positive of the following statement: Conditional: If two angles form a linear pair, then they are supplementary. Converse: Inverse: Contra – positive:

  19. DOGS A =poodle ... a dog ...B  dog .A B= horse ... NOT a dog . B Venn diagrams: • show relationships between different sets of data. • can represent conditional statements. • is usually drawn as a circle. • Every point IN the circle belongs to that set. • Every point OUT of the circle does not. Lesson 2-2: Logic Example:

  20. For all..., every..., if...then... All right angles are congruent. Example1: Congruent Angles Example 2: Every rose is a flower. Right Angles Flower • lines that do • not intersect Lesson 2-2: Logic Rose parallel lines Example 3: If two lines are parallel, then they do not intersect.

  21. To Show Relationships using Venn Diagrams: Blue or Brown (includes Purple) … AB A B A  B Lesson 2-2: Logic

  22. The Venn Diagram shows the number of students enrolled in Moniques’ dance school for tap, jazz and ballet classes • How Many students are in all three classes? • How many in tap or ballet? • How many are in jazz and ballet but not tap? Tap Jazz 13 28 43 9 17 25 29 Ballet