#### Math - 2017-18

# G.1 and *G.1 - Logic and Proofs

**G.1** The student will **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) using Venn diagrams to represent set relationships; and

d) using deductive reasoning.

G.1The student willusedeductive reasoning toconstructandjudgethe validity of a logical argument consisting of a set of premises and a conclusion. This will includea)

identifyingthe converse, inverse, and contrapositive of a conditional statement;b)

translatinga short verbal argument into symbolic form;c)

determiningthe validity of a logical argument.

**Bloom's Level:** Evaluate

*Adopted: 2009*

### 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

- 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.
- 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, and theorems.
- Logical arguments consist of a set of premises
or hypotheses and a conclusion.
- Euclidean geometry is an axiomatic system based
on undefined terms (point, line and plane), postulates, and theorems.
- When a conditional and its converse are true,
the statements can be written as a biconditional, i.e.,
*iff*or*if and only if*. - Logical arguments that are valid may not be
true. Truth and validity are not
synonymous.

· 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

G.1a_{1}
**Identify** the converse,
inverse, and contrapositive of a condition statement.

· G.1a_{1} **Identify**
the converse, inverse, and contrapositive of a conditional statement.

G.1b_{1} **Translate** verbal arguments into
symbolic form, such as (*p* → *q*) and (~*p* →
~*q*).

· G.1b_{1} **Translate**
verbal arguments into symbolic form using the symbols of formal logic.

G.1c_{2}
**Interpret** Venn diagrams.

G.1d_{1}
**Determine** the validity of a
logical argument.

· G.1c_{1} **Determine**
the validity of a logical argument using valid forms of deductive reasoning.

G.1b_{2}
**Recognize** and **use** the symbols of formal logic, which
include

G.1c_{1}
**Use** Venn diagrams to **represent** relationships, such as
intersection and union.

G.1d_{2} **Use**
valid forms of deductive reasoning, including the law of syllogism, the law of contrapositive,
the law of detachment, and counterexamples.

· G.1c_{2} **Determine**
that an argument is false using a counterexample.

G.1d_{3}
**Select** and **use** various types of reasoning and methods
of proof as appropriate.

### 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: Oct 27, 2017*