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
STATEMENT: A declarative sentence that is either TRUE
or FALSE (but not both at once).
Examples of statements
I ate spinach for lunch today.
5 + 8 = 10
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).
Examples of statements
I ate spinach for lunch today.
5 + 8 = 10
The following are NOT statements
Hit the baseball
Are you listening?
If I were a rich man
This sentence is false.
(This last one is called a paradox because if we assume it is true, it has to be false
and 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.).
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
Dickens wrote novels and the poem is short.
You can pay me now or you can pay me later.
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.
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.
Examples
Statement: "Earth is a planet.“
Negation: "Earth is not a planet."
Statement: "Today is not Tuesday."
Negation: "Today is Tuesday."
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.
Examples
Statement: "Earth is a planet.“
Negation: "Earth is not a planet."
Statement: "Today is not Tuesday."
Negation: "Today is Tuesday."
What is the negation of the statement: "x < 7 " ?
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.
Examples
Statement: "Earth is a planet.“
Negation: "Earth is not a planet."
Statement: "Today is not Tuesday."
Negation: "Today is Tuesday."
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
MATH 110 Sec 3-1 Lecture on Statements and Connectives
Symbols for our connectives
Connective
Symbol
Name
and
⋀
Conjunction
MATH 110 Sec 3-1 Lecture on Statements and Connectives
Symbols for our connectives
Connective
Symbol
Name
and
⋀
Conjunction
or
⋁
Disjunction
(inclusive)
MATH 110 Sec 3-1 Lecture on Statements and Connectives
Symbols for our connectives
Connective
Symbol
Name
and
⋀
Conjunction
or
⋁
Disjunction
∽
Negation
(inclusive)
not
MATH 110 Sec 3-1 Lecture on Statements and Connectives
Symbols for our connectives
Connective
Symbol
Name
and
⋀
Conjunction
or
⋁
Disjunction
∽
→
Negation
Implication
(inclusive)
not
If…then
MATH 110 Sec 3-1 Lecture on Statements and Connectives
Symbols for our connectives
Connective
Symbol
Name
and
⋀
Conjunction
or
⋁
Disjunction
∽
→
↔
Negation
Implication
Biconditional
(inclusive)
not
If…then
If and only if
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."
SYMBOLIC
STATEMENT
STATEMENT IN ENGLISH
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."
SYMBOLIC
STATEMENT
𝑝∧𝑞
STATEMENT IN ENGLISH
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."
SYMBOLIC
STATEMENT
STATEMENT IN ENGLISH
𝑝∧𝑞
The car is blue and the dog is big.
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."
SYMBOLIC
STATEMENT
𝑝∧𝑞
∼𝑝
STATEMENT IN ENGLISH
The car is blue and the dog is big.
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."
SYMBOLIC
STATEMENT
STATEMENT IN ENGLISH
𝑝∧𝑞
∼𝑝
The car is blue and the dog is big.
The car is not blue.
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."
SYMBOLIC
STATEMENT
𝑝∧𝑞
∼𝑝
𝑝∨𝑞
STATEMENT IN ENGLISH
The car is blue and the dog is big.
The car is not blue.
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."
SYMBOLIC
STATEMENT
STATEMENT IN ENGLISH
𝑝∧𝑞
∼𝑝
𝑝∨𝑞
The car is blue and the dog is big.
The car is not blue.
The car is blue or the dog is big.
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."
SYMBOLIC
STATEMENT
𝑝∧𝑞
∼𝑝
𝑝∨𝑞
𝑝 ∧∼ 𝑞
STATEMENT IN ENGLISH
The car is blue and the dog is big.
The car is not blue.
The car is blue or the dog is big.
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."
SYMBOLIC
STATEMENT
STATEMENT IN ENGLISH
𝑝∧𝑞
∼𝑝
𝑝∨𝑞
𝑝 ∧∼ 𝑞
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 and the dog is not big.
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."
SYMBOLIC
STATEMENT
𝑝∧𝑞
∼𝑝
𝑝∨𝑞
𝑝 ∧∼ 𝑞
𝑝 →∼ 𝑞
STATEMENT IN ENGLISH
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 and the dog is not big.
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."
SYMBOLIC
STATEMENT
STATEMENT IN ENGLISH
𝑝∧𝑞
∼𝑝
𝑝∨𝑞
𝑝 ∧∼ 𝑞
𝑝 →∼ 𝑞
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 and the dog is not big.
If the car is blue, then the dog is not big.
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."
SYMBOLIC
STATEMENT
𝑝∧𝑞
∼𝑝
𝑝∨𝑞
𝑝 ∧∼ 𝑞
𝑝 →∼ 𝑞
∼𝑞↔𝑝
STATEMENT IN ENGLISH
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 and the dog is not big.
If the car is blue, then the dog is not big.
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."
SYMBOLIC
STATEMENT
STATEMENT IN ENGLISH
𝑝∧𝑞
The car is blue and the dog is big.
∼𝑝
The car is not blue.
𝑝∨𝑞
The car is blue or the dog is big.
𝑝 ∧∼ 𝑞
The car is blue and the dog is not big.
𝑝 →∼ 𝑞
If the car is blue, then the dog is not big.
∼ 𝑞 ↔ 𝑝 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
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)
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.
None of the drivers was hurt.
Each student made an A.
No joke was funny.
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.
None of the drivers was hurt.
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)
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.
None of the drivers was hurt.
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)
At least one score was a touchdown.
Some kids are kind.
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.
Sometimes we are asked to find the negation of a statement
with a universal or an existential quantifier.
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.
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.
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.
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.
STATEMENT
NEGATION
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.
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.
STATEMENT
All athletes are wealthy.
NEGATION
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.
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.
STATEMENT
NEGATION
All athletes are wealthy.
Some athletes are not wealthy.
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.
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.
STATEMENT
NEGATION
All athletes are wealthy.
Some students get loans.
Some athletes are not wealthy.
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.
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.
STATEMENT
NEGATION
All athletes are wealthy.
Some students get loans.
Some athletes are not wealthy.
No student gets a loan.