pure mathematics- reasoning and logic

Simple statementSimple statement

A mathematical sentence is a sentence

that states a fact or contains a complete idea. A sentence that

can be judged to be true or false is called a

statement , or a closed sentence.

Example:

St1: Jill has maintained a GPA of 2.5 or better.

St2: Jill has missed no more than two

practices

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Truth value of a Truth value of a statementstatement

It is either true or false

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Truth table Truth table

Is the way to show the truth value of a compound statement

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Compound statementCompound statement

It is formed by combining two or more simple statements.

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Symbolic LogicSymbolic Logic

Connective Symbol Formal Name

Not ~ negation

And ^ conjunction

or v disjunction

If… then → conditional

…if and only if ↔ Bi-conditional

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Symbolic Logic in Symbolic Logic in compound sentencecompound sentence

Connective Symbol Formal Name Compound Statement using connective

Compound Statement using symbols

Not ~ negation

And ^ conjunction P and Q P ^ Q

or ν disjunction

If… then → conditional

…if and only if ↔ Bi-conditional

P: represents a statement( ST1)Q: represents other statement( ST2) They can be true or false

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TruthTruth table table for conjuncti for conjunctionon

•P(ST1): Jill has maintained a GPA of 2.5 o better

•Q(ST2): Jill has missed no more than two practices.

Example P Q P and Q

Jill has a 3.2 GPA and has missed 1 practice

True True True

Jill has a GPA and has missed 3 practices

True False false

Jill has a 2.25 GPA and has missed no practices

False True False

Jill has a 2.4 GPA and has missed 3 practices

False False False

Note: - A conjunction is true only when all of its simple statements are true. - If any statement is false , the conjunction is false

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Truth Truth tabletable for disjunctio for disjunction ( P or Q)n ( P or Q)

P: “ It is weekend”

Q: “It is holiday”Example P Q P or Q

It is SaturdayIt is a holiday

True True True

It is SundayIt is not holiday

True False True

It is Wednesday It is a holiday

False True True

It is TuesdayIt is not a holiday

False False False

Note:- A disjunction is true when any of its simple statements are true. - The disjunction is false only when all of its statements are false.

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NegationNegation

Indicates the opposite, usually employing the word not.

St3: Today is monday.

Negation of the St3:Today is not monday.

St4:That was fun.

Negation of the St4: That was not fun.

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Conditional Conditional StatementsStatementsSimple Sentences Compound Sentence : Conditional (p →q )

p: You are absent q: You have a make up assignment to complete

If you are absent, then you have a make up assignment to complete.

P: Pigs flyQ: I’ll go to the dentist

If pigs fly, then I’ll go to the dentist.

Conective: If ...then....Conective: If ...then....

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Truth TableTruth TableExample p q p→ q

You are lateYou are on time

True False False

A human is a catSquares have corners

False True True

Note: The only way that the conditional is a false statement is when a true ‘if’ clause leads to a false ‘then’ clause( T→F)

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Bi- ConditionalBi- Conditional

A bi-conditional statement is defined to be true whenever both parts have the same

truth value.

Symbol: ↔

Connective: ...if and only if (iff)

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Bi- Conditional Bi- Conditional StatementStatementSimple Sentences Compound Sentence : Bi- Conditional ( p ↔q)

p: A polygon is a triangle( T)q: A polygon has exactly 3

sides( T)

“ A polygon is a triangle if and only if it has exactly 3 sides”(T)

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Bi- Conditional Truth Bi- Conditional Truth TableTable

p q P↔ q

T T T

T F F

F T F

F F T

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General Truth TableGeneral Truth Table p q p v q p ^ q p → q p↔ q

T T T T T T

T F T F F F

F T T F T F

F F F F T T

• A conditional statement conditional statement consists of two parts, a hypothesis in the “if” clause and a conclusion in the “then” clause. For instance, “If it rains, then they will cancel school.”

• “It rains, “is the hypothesis.

• “They will cancel school,” is the conclusion.

• To form the converse of the conditional statement, interchange the hypothesis and the conclusion.

• The converse of “If it rains, then they will cancel school” is “If they cancel school, then it rains.”

• To form the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion.

• The inverse of “If it rains, then they will cancel school” is “If it does not rain, then they do not cancel school.”

• To form the contrapositive of the conditional statement, interchange the hypothesis and the conclusion of the inverse statement.

• The contrapositive of “If it rains , then they will cancel school is “If they do not cancel school then it does not rain

Symbolic representation.

Statement If p, then q.

Converse If q, then p.

Inverse If not p, then not q.

Contrapositive If not q, then not p.

If the statement is true, then the contrapositive is also logically true. If the converse is true, then the inverse is also logically true.

Statement If two angles are congruent, then they have the same measure.

Converse If two angles have the same measure, then they are congruent.

Inverse If two angles are not congruent, then they do not have the same measure.

Contrapositive If two angles do not have the same measure, then they are not congruent.

Logical Equivalence

A logical equivalence is a statement that two mathematical sentence forms are completely interchangeable:if one is true, so is the other; if one is false, so is the other.

For example, we could express that an implication is equivalent to its contrapositive in either of the following ways: A → B is (logically) equivalent to (not B) → ( not A)

or

‘’ ( A → B ) ↔( not B) → ( not A ) is a logical equivalence

De Morgan's Laws

• not( A or B) →( not A) and ( not B)

• not( A and B) →( not A) or ( not B)

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