MATH 110 Sec 3-1 Lecture on Statements and Connectives STATEMENT: A declarative sentence that is...

MATH 110 Sec 3-1 Lecture on Statements and Connectives STATEMENT: A declarative sentence that is either TRUE

or FALSE (but not both at once).

MATH 110 Sec 3-1 Lecture on Statements and Connectives

Examples of statementsI ate spinach for lunch today.

STATEMENT: A declarative sentence that is either TRUE or FALSE (but not both at once).



5 + 8 = 10




5 + 8 = 10The following are NOT statements





Hit the baseball





Hit the baseballAre you listening?






If I were a rich man






If I were a rich manThis sentence is false.






If I were a rich manThis sentence is false.

(This last one is called a paradox because if we assume it is true, it has to be falseand if we assume that it is false, then it has to be true.)


MATH 110 Sec 3-1 Lecture on Statements and Connectives COMPOUND STATEMENTS

are formed by 'combining' simple statements using various connectives ('and', 'or', 'not', 'if...then', etc.).



Examples of COMPOUND statements



Examples of COMPOUND statementsDickens wrote novels and the poem is short.



Examples of COMPOUND statementsDickens wrote novels and the poem is short.You can pay me now or you can pay me later.




Today is not payday.




Today is not payday.If today is Wednesday, then I have math class.

MATH 110 Sec 3-1 Lecture on Statements and Connectives The NEGATION of a statement is often formed using the word

'not‘.The negation of a statement always has the opposite truth value of the original statement.


'not‘.

ExamplesThe negation of a statement always has the opposite truth value of the original statement.


'not‘.

ExamplesStatement: "Earth is a planet.“

The negation of a statement always has the opposite truth value of the original statement.


'not‘.


Negation: "Earth is not a planet."



'not‘.


Negation: "Earth is not a planet."Statement: "Today is not Tuesday."



'not‘.



Negation: "Today is Tuesday."



'not‘.





What is the negation of the statement: "x < 7 " ?


'not‘.





What is the negation of the statement: "x < 7 " ?x > 7

MATH 110 Sec 3-1 Lecture on Statements and Connectives Symbols for our connectives

Connective Symbol Name

and Conjunction



and Conjunction

or

Disjunction



and Conjunction

or (inclusive)

Disjunction



and Conjunction

or (inclusive)

Disjunction

not Negation



and Conjunction

or (inclusive)

Disjunction

not NegationIf…then Implication



and Conjunction

or (inclusive)

Disjunction

not NegationIf…then Implication

If and only if Biconditional

MATH 110 Sec 3-1 Lecture on Statements and Connectives Let p represent "The car is blue." and let q represent "The dog is big."

SYMBOLICSTATEMENT STATEMENT IN ENGLISH



The car is blue and the dog is big.



The car is blue and the dog is big.The car is not blue.




The car is blue or the dog is big.




The car is blue or the dog is big.The car is blue and the dog is not big.





If the car is blue, then the dog is not big.





If the car is blue, then the dog is not big.The dog is not big if and only if the car is blue.

MATH 110 Sec 3-1 Lecture on Statements and Connectives Universal and Existential Quantifiers


‘All’, ‘Each’, ‘Every’, ‘No’, ‘None’ are universal quantifiers.(asserts that the statement quantified is true for EVERY relevant thing)


All dogs bark.



All dogs bark. None of the drivers was hurt.



All dogs bark.Each student made an A.

None of the drivers was hurt.



All dogs bark.Each student made an A. No joke was funny.





‘Some’, ‘There exists’, ‘At least one’ are existential quantifiers. (asserts that the statement quantified is true for AT LEAST ONE THING)




Some kids are kind.






Some kids are kind. At least one score was a touchdown.






Some kids are kind. At least one score was a touchdown.There exists a UA student who doesn’t like football.





MATH 110 Sec 3-1 Lecture on Statements and Connectives ‘All’, ‘Each’, ‘Every’, ‘No’, ‘None’ are universal quantifiers.

‘Some’, ‘There exists’, ‘At least one’ are existential quantifiers.

MATH 110 Sec 3-1 Lecture on Statements and Connectives ‘All’, ‘Each’, ‘Every’, ‘No’, ‘None’ are universal quantifiers.

‘Some’, ‘There exists’, ‘At least one’ are existential quantifiers.

Sometimes we are asked to find the negation of a statement with a universal or an existential quantifier.


Sometimes we are asked to find the negation of a statement with a universal or an existential quantifier. Helpful Hint: The negation of a Universal statement will be an Existential statement

AND the negation of an Existential statement will be a Universal statement.

‘All’, ‘Each’, ‘Every’, ‘No’, ‘None’ are universal quantifiers.‘Some’, ‘There exists’, ‘At least one’ are existential quantifiers.




STATEMENT NEGATION





STATEMENT NEGATION

All athletes are wealthy.





STATEMENT NEGATION

All athletes are wealthy. Some athletes are not wealthy.





STATEMENT NEGATION

All athletes are wealthy. Some athletes are not wealthy.Some students get loans.





STATEMENT NEGATION

All athletes are wealthy. Some athletes are not wealthy.Some students get loans. No student gets a loan.


MATH 110 Sec 3-1 Lecture on Statements and Connectives STATEMENT: A declarative sentence that is...

Documents

Transcript of MATH 110 Sec 3-1 Lecture on Statements and Connectives STATEMENT: A declarative sentence that is...