LOGIC, CONDITIONAL STATEMENTS AND DEDUCTIVE REASONINGSol: G.1 b, c

Sec: 2.1-2.2

A conjecture is an educated guess based on known information.

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.

NEGATIONS

The negation of a statement says it has the opposite meaning.

So the negation would be ~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.

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.

Two types – conjunction and disjunction

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.*

andqp

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 daysr: A plane is defined by 3 non-collinear points.

a) p and qOne 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 non-collinear points.

pr

rq ~

rp ~

WRITE A COMPOUND STATEMENT FOR THE CONJUNCTION AND FIND ITS TRUTH VALUE.

p: an elephant is a mammalq: a square has four right angles

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.*

qp

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.

AB

AB

:rq

WRITE A COMPOUND STATEMENT FOR THE DISJUNCTION AND FIND ITS TRUTH VALUE.

p: a diameter of a circle is twice its radiusq: a rectangle has four equal sides.

12

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.

Example: A =poodle ... a dog

B= horse ... NOT a dog

.BDOGS

.A...B dog

Animals

Dogs Cats

Polygons

Triangles Squares

Flowers

Red Roses

Hamburgers

With With Cheese Onions

17

All right angles are congruent. Congruent Angles

Right Angles

Example1:

18

Example 2: Every rose is a flower.

Flower

Rose

Lesson 2-2: Logic 19

TO SHOW RELATIONSHIPS USING VENN DIAGRAMS:

A B

A B

THE VENN DIAGRAM SHOWS THE NUMBER OF STUDENTS ENROLLED IN MONIQUES’ DANCE SCHOOL FOR TAP, JAZZ AND BALLET CLASSES

a) How Many students are in all three classes?b) How many in tap or ballet?c) How many are in jazz and ballet but not

tap?

Tap Jazz

Ballet

28

43

29

917

25

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CONDITIONAL STATEMENT

Definition:

A conditional statement is a statement that can be written in if-then form.“If _____________, then ______________.”

“if p, then q”. Symbolic Notation p → q

CONDITIONAL STATEMENT

Conditional Statements have two parts:

The hypothesis is the part of a conditional statement that follows “if” (Usually denoted p.)

The conclusion is the part of an if-then statement that follows “then” (Usually denoted q.)

The hypothesis is the given information, or the condition.

The conclusion is the result of the given information.

EXAMPLEWrite 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.

EXAMPLEIdentify the Hypothesis and Conclusion in the

following statements:

1. If a polynomial has six sides, then it is a hexagon.

H: A polygon has 6 sides C: it is a hexagon

2. 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

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.

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.

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.

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.

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.

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:

ASSIGNMENTS

Classwork: Handout

Homework: pg 78-79 16-26 even and 40-44 even

DEDUCTIVE REASONING

Definition – Use facts, definitions, accepted proportions and the laws of logic to form a logical argument.

There are two laws of logic:1. The law of Detachment2. The law of Syllogism

LAW OF DETACHMENT

Definition – If the Hypothesis of a true conditional statement is true, then the conclusion is also true. So, if p → q is true and p is true, the q is true.

Symbolic Representation: p → q p

Ex: Angles that are supplementary have measures

with a sum of 180°. < A and < B are Supplementary < A and < B measures are a sum of 180°

q

LAW OF SYLLOGISM

Definition – If hypothesis p, then conclusion q.If hypothesis q , then conclusion r.(if both above statements are true) If hypothesis p, then conclusion r.(Then the above is also true)Symbolic Representation: p → q

q → r

Ex: The sun is a star Stars are in constant motion. The sun is in constant motion.

rp

Determine if statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it does not, write invalid.

In – line skates live dangerously. If you live dangerously, then you like to dance. If you are an in – line skater, then you like to dance.

If you drive safely, the life you save may be your own. Shani drives safely. The life she saves may be her own.

  If a figure is a rectangle, then its opposite sides are congruent. AB DC and AD BC. ABCD is a rectangle.

Determine if a valid conclusion can be reached from the two true statements using the Law of Detachment or the Law of Syllogism. If a valid conclusion is possible, state it and the law that is used. If a valid conclusion does not follow, write no conclusion.

If two angles are vertical, then they do not form a linear pair.

If two angles are vertical, then they are congruent.   If you eat to live, then you live to eat. Odina eats to live.   Cars are useful. Useful cars are practical. 

ASSIGNMENTS

Classwork: WorksheetHomework: Pg 84-85 8-9, 24-29