Download MATH 311W Wksht 1 • A logical statement is a phrase that is

MATH 311W Wksht 1
• A logical statement is a phrase that is unambiguously true or false. By unambiguous,
we mean that the truth or falsehood of the statement is not a matter of opinion or
interpretation and is universally accepted.
• Which of the following are logical statements?:
“It is raining.”
“2 + 2 = 9”
“Sunday was a nice day.”
“5 is a lovely number.”
• Every/All Statements. An“every/all” statement is true only if it is true in all cases,
or else the statement is false. In other words, it only takes one counterexample to prove
an “every” statement false.
Explain why the following are false statements:
“Every month has at least 30 days.”
“Everyone loves ice cream.”
“All prime numbers are odd.”
• Some/There exists Statements. A “some” statement takes only one example to
prove it is true. A “some” statement is false if there are no examples that satisfy the
statement.
Which of the statements are true?
“Some people live to be 100.”
“There exist prime numbers that are even.”
“Some numbers are equal to their squares.”
“Some numbers are odd and divisible by 2.”
NOTE on Grammar: “There exist prime numbers that are even” is true even though
there is only ONE even prime number. The plural ’prime numbers’ doesn’t require
us to find more than one even prime number. However, the statement is more clearly
written as “Some prime numbers are even.”
• “Not” Statements/Negation. If A is a statement, then its negation (labelled as
∼ A) or “not A” is also a statement. If A is true then ∼ A is false and if A is false
then ∼ A is true.
Negate the following:
A: “It is raining.”
∼ A:
A: “2 + 2 equals 9.”
∼ A:
A: “Every prime number is odd.”
∼ A:
Sometimes it is helpful when negating a statement to write “It is not true that ...”
before the statement.
For example,
A :“Not every number is odd”
∼ A =“It is not true that not every number is odd”
• “Not every” statements Any “not every” statement can be made into an equivalent
“some are not” statement:
“Not every prime number is odd” is equivalent to the statement “some prime number
is not odd.”
“Not everyone likes ice cream” is equivalent to “Someone does not like ice cream”
“Not every prime is divisible by 1” is equivalent to “Some prime number is not divisible
by 1” (Both of these statements are false)
“Not every person is here” is equivalent to:
• “Not some/ do not exist” statements Any “not some” statement can be made
into an equivalent “every are not” statement
“It is not true that some numbers that have Property X” is equivalent to the statement
“Every number does not have Property X.”
“It is not true that someone likes ice cream” is equivalent to “Everyone does not like
ice cream.”
“There are not some rectangles that are squares” is equivalent to:
• Two statements A and B are equivalent if whenever A is true, then B is also true
and whenever A is false then B is also false, and vice versa. We write A ⇔ B when A
is equivalent to B.
(e.g.) ∼ (∼ A) ⇔ A (double negative)
• And/Or Statements. If A is a statement and B is a statement, then the compound
statement: “A and B” is a new statement. “A and B” is true only if BOTH A and B
are true. We usually write this statement as A ∧ B.
“A or B” is also a compound statement that is true if at least one of A or B are true
(or possibly both). We usually write this statement as A ∨ B
(a) ”It is raining and Jennifer is awake” is true exactly when both statements are true.
(b) ”It is raining or Jennifer is awake” is true if either it is raining, or if Jennifer is
awake, or both.
(c) ”It is raining and it is not raining” can never be true (why)?
(d) ”It is raining or it is not raining” is always true (why)?
NOTE: The mathematical use of the word OR is not the same as the common English
(exclusive) use of the word. Meaning in English, we usually use “or” to mean not
both. For example in everyday English, “He will have Coke or Pepsi” usually does not
include the possibility of having both, but in mathematics: the following statements
are true:
The number 7 is positive or prime. (True, it is both).
The number 2 is even or prime. (True, it is both).
• Truth Tables. The truth or falsehood of a compound statement that depends on
other statements can be determine using a truth table.
For example the (variable) statement: A ∨ B can be determined based on whether A
or B are true (there are 4 possibilities (see top of pg 10 in N&B):
A
T
T
F
F
B A∨B A∧B
T
T
T
F
T
F
T
T
F
F
F
F
One reads the across a row of the truth table for a particular possibility for the statements A and B (in the last, we have the case where both A and B are assumed false).
• Negating And/Or Statements.
The statement “not (A and B)” ∼ (A ∨ B) is the same thing as “(not A) or (not B).”
[Notice that without the parentheses this is ambiguous!].
Negate the compound statement:
A: ”It is cloudy and it is Friday”
Then ∼ A is equivalent to: ”It is not cloudy or not Friday”.
NOTE that in English, the actual statement of ∼ A is ”It is not true that: it is cloudy
and it is Friday” which is awkward and complicated.
In fact, no matter what A and B are (true or not) it is always fact that ∼ (A ∧ B) ⇔
(∼ A) ∨ (∼ B). We prove this with a truth table:
• Theorem 1 ∼ (A ∨ B) ⇔ (∼ A) ∧ (∼ B)
Proof
A
T
T
F
F
B A ∨ B ∼ (A ∨ B) (∼ A) ∧ (∼ B)
T
T
F
F
F
T
F
F
T
T
F
F
F
F
T
T
As you can see, these two statements match (are simultaneously true or false) under
all possible cases, so they are equivalent.
NOTE: In logic, we care a lot about when two statements are equivalent, the statements
themselves do not always have to be true. When we prove theorems, we will care about
statements being true.
HW # 1: [Read N&B: pg 2 (definition of statement), and pg 25-27 look at Ex. 1-10,
pg 30]
1. Explain whether the phrases below are logical statements. Explain.
(a) This class is hard.
(b) The number 4 is odd.
(c) My car is a smooth ride.
(d) George Washington wore false teeth.
2. Negate each of the following statements:
(a) Tom ate his broccoli and his cauliflower.
(b) 7 is an odd number.
(c) Every whole number is even.
(d) Some number is bigger than 10 but less than 12.
(e) Tom is taller than Sally, but Jane is the smallest.
3. Prove the following using a truth table:
Theorem 2 ∼ (A ∧ B) ⇔ (∼ A) ∨ (∼ B)
4. (Challenge) If possible, find a compound statement involving A and B that works
with the following truth table.
A
T
T
F
F
B Your compound statement
T
F
F
F
T
T
F
T