Statements, Connectives, &
Quantifiers
MATH 102
Contemporary Math
S. Rook
Overview
• Section 3.1 in the textbook:
– Statements
– Connectives
– Quantifiers
Statements
Statements
• A statement (logic-wise) is a declarative sentence
(i.e. one that is either true or false)
– Represented symbolically by using lowercase letters
– Concerned only if it is possible for the sentence to
evaluates to EITHER true OR false
• Statements CANNOT be exclamations or
questions
– See sentences e) – g) on pg 83 of the textbook
• What are examples of statements? There are
hardly any wrong answers!
Simple & Compound Statements
• A simple statement contains ONE idea
• Negating a statement, denoted as ~, means to write
the opposite of the original statement
– e.g. Negate one of the statements we discussed previously
• A compound statement contains more than one idea
by joining simple statements together using
connectives (bridges)
• Four categories of connectives – we will discuss each in
greater detail in a few slides:
– Conjunction & Disjunction
– Conditional & Biconditional
Negation of Statements
Ex 1: Write the negation of the statement
symbolically:
a) I forgot to feed the cat today.
b) He does not cheat at cards.
Connectives
Conjunction & Disjunction
• A conjunction, symbolized by  , is a compound
statement that uses the word and to connect
statements
– Why is a conjunction a compound statement?
– e.g. Take two additional statements from the “pool”
and form a conjunction
• A disjunction, symbolized by  , is a compound
statement that uses the word or to connect
statements
– e.g. Take two additional statements from the “pool”
and form a disjunction
Conjunction & Disjunction (Example)
Ex 2: Consider the following statements.
p: He can juggle.
q: I know how to speak German.
a) Express symbolically: He can not juggle
and I know how to speak German.
b) Write in English: ~p v ~q
Conditional & Biconditional
• A conditional, symbolized by  , is a compound
statement that connects two statements in an
if …, then structure
– e.g. Take two additional statements from the “pool”
and form a conditional
• A biconditional, symbolized by  , is a
compound statement that connects two
statements in an if and only if structure
– e.g. Take two additional statements from the “pool”
and form a biconditional
Conditional & Biconditional (Example)
Ex 3: Consider the following statements.
p: The defendant is convicted of perjury.
q: He will spend at least 30 years in jail.
a) Express symbolically: If the defendant is
convicted of perjury, then he will spend less
than 30 years in jail.
b) Write in English: ~ p  ~ q
Quantifiers
Quantifiers
• Quantifiers are words or phrases in a statement that
answer the question “how many?”
• Two common classes of quantifiers:
– Universal: a phrase that indicates EVERY object
satisfies a given property
• e.g. All, every
• e.g. Take a statement from the “pool” and modify it to use a
universal quantifier
– Existential: a phrase that indicates ONE OR MORE
objects satisfy a given property
• e.g. Some, there exists, there is at least one
• e.g. Take a statement from the “pool” and modify it to use
an existential quantifier
Negating a Universal Quantifier
• Consider negating the following statement:
Every student in this room will get a ‘C.’
– What does it mean for this statement to be false?
– How would we negate the statement?
• The negation of a universal quantifier is an
existential quantifier
Negating an Existential Quantifier
• Consider negating the following statement:
Some students use a calculator.
– What does it mean for this statement to be false?
– How would we negate the statement?
– What is another, more concise, way to write this?
• The negation of an existential quantifier is a
universal quantifier
Negating Quantifiers (Example)
Ex 4: Write the negation of each statement in
English:
a) All exams in this class require studying.
b) Some professional wrestling matches are not
scripted.
c) At least one item on the McDonald’s menu is a
healthy choice.
Summary
• After studying these slides, you should know how to do
the following:
– Identify statements and differentiate between simple and
compound statements
– Negate a statement
– Write simple and compound statements symbolically
– Understand the meaning of quantifiers
– Negate statements that contain quantifiers
• Additional Practice:
– See the list of suggested problems for 3.1
• Next Lesson:
– Truth Tables (Section 3.2)