Inductive Reasoning and Conjecture, and Logic

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Presentation transcript:

Inductive Reasoning and Conjecture, and Logic

Inductive Reasoning Reasoning that uses examples to come to a conclusion; think patterns A concluding statement based on inductive reasoning is called a conjecture; think educated guess

Examples The sum of two even numbers The relationship between AB and EF if AB = CD and CD = EF

Counterexamples A false example; it can be a number, a drawing, or a statement. Ex. – 2x > x; false if x < 0

Examples 3, 3, 6, 9, 15,… what is the next number? What is next in the sequence? If in ∆ABC, (AB)2 + (BC)2 = (AC)2, then ∆ABC is a right triangle

Logic Statement is a sentence that is either true or false. Truth value of a statement is either true (T) or false (F) Negation of a statement has the opposite meaning and truth value; written p, read not p

Logic A compound statement using the word and is called a conjunction; written , read p and q Conjunctions are only true when both statements are true

Logic A compound statement using the word or is called a disjunction; written , read p or q Disjunctions are only false when both statements are false

Truth Table Truth tables organize the values of statements Start by making columns with heading that include each original statement, any negations of the statements, and the compound statements Next, determine all of the possible combinations of truth values… 2x, where x is the number of original statements

Truth Table Use the truth values to determine each part of the compound statement to determine the truth value of the statement.

Venn Diagrams Conjunctions and disjunctions can also be illustrated with Venn diagrams. Rectangles are located at the intersection of the set of quadrilaterals and the set of convex polygons

Venn Diagrams A disjunction is represented by the union of the two sets; all polygons that are quadrilaterals, convex or both. The disjunction includes three regions: quadrilaterals that are not convex, convex polygons that are not quadrilaterals, and polygons that are both quadrilaterals and convex

Examples

Examples

Examples

Questions How are truth tables useful in the real world? How can you as a student use Venn diagrams in your life?