# G.1 - Logic and Proofs

G.1  The student will use deductive reasoning to construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. This will include

a)  identifying the converse, inverse, and contrapositive of a conditional statement;

b)  translating a short verbal argument into symbolic form;

c)  determining the validity of a logical argument.

### BIG IDEAS

• I can analyze the logic behind advertising, decide the official weight of a Hershey bar by weighing a fraction of the candy bars coming off a production line, and present a winning a court case.
• I will develop logical critical thinking by drawing conclusions and making inferences from known or assumed facts.

### UNDERSTANDING THE STANDARD

2016 VDOE Curriculum Framework - G.1 Understanding

·  Inductive reasoning, deductive reasoning, and proof are critical in establishing general claims.

·  Deductive reasoning is the method that uses logic to draw conclusions based on definitions, postulates, and theorems.

·  Valid forms of deductive reasoning include the law of syllogism, the law of contrapositive, the law of detachment, and the identification of a counterexample.

·  Symbolic notation is used to represent logical arguments, including the use of , , , , , and .

·  The law of syllogism states that if p q is true and q  r is true, then
p r is true.

·  The law of contrapositive states that if p q is true and q is true, then p is true.

·  The law of detachment states that if p q is true and p is true, then
q is true.

·  A counterexample is used to show an argument is false.

·  Inductive reasoning is the method of drawing conclusions from a limited set of observations.

·  Proof is a justification that is logically valid and based on initial assumptions, definitions, postulates, theorems, and/or properties.

·  Logical arguments consist of a set of premises or hypotheses and a conclusion.

·  When a conditional (p q) and its converse (q p) are true, the statements can be written as a biconditional, p iff q; or p if and only if q;or p  q.

·  Logical arguments that are valid may not be true.  Truth and validity are not synonymous.

·  Exploration of the representation of conditional statements using Venn diagrams may assist in deepening student understanding.

·  Formal proofs utilize symbols of formal logic to determine validity of a logical argument.

### ESSENTIALS

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

·  G.1a1  Identify the converse, inverse, and contrapositive of a conditional statement.

·  G.1b1  Translate verbal arguments into symbolic form using the symbols of formal logic.

·  G.1c1  Determine the validity of a logical argument using valid forms of deductive reasoning.

·  G.1c2  Determine that an argument is false using a counterexample.

### KEY VOCABULARY

conditional statement, converse, inverse, contrapositive, verbal argument, hypothesis/premises, conclusion, symbolic form, deductive reasoning,  inductive reasoning, law of syllogism, law of contrapositive, law of detachment, counterexample, proof, Venn diagram , intersections, unions, negation, therefore,  compound, conjunction, disjunction, conjecture, truth value, validity, truth table, biconditional, postulates/axioms, theorems, Euclidean geometry, axiomatic system, undefined terms, point, line, plane, algebraic properties

Updated: Jul 30, 2019