MATH 111 Textbook: Epp, Susanna S', Discrete Mathematics with Applications Instructor: Cansu Betin

1
MATH 111Textbook Epp, Susanna S., Discrete
Mathematics with Applications Instructor Cansu
Betin
1.1 LOGICAL FORM AND LOGICAL EQUIVALENCES AND
1.2 CONDITIONAL STATEMENTS

• Definition A statement (or a Proposition) is a
  sentence that is TRUE or FALSE but not
  ambiguous (not both).
• Examples
• Snow falls in winter (TRUE statement)
• Two plus two equals four (TRUE statement)
• Two plus two equals five (FALSE statement)
• I have 17 children (FALSE statement)
• He is 6 feet tall (NOT a statement, ambiguous
  He ?)
• When will you go to Izmir? (NOT a statement)
• Happy new year (NOT a statement)
• Ankara is the capital city of Turkey (TRUE
  Statement)
• What a beatiful day! (NOT a statement)
• Dogs have four feet (TRUE Statement)
• 235 (TRUE Statement)
• 3lt5-2 (FALSE Statement)

2

• Def A statement which can not be separated into
  simpler statements is called primitive
  statements.
• p Peter hates Lewis
• q v2 gt 1
• r It is raining outside
• p, q and r are primitive statements.
• Logical Connectives
• (or ) not (negation)
• ? and (conjunction)
• ? or (disjunction)
• ? imply (if then) (conditionals)
• ? if and only if (equivalent) (biconditionals)
• Using logical connectives we obtain compound
  statements.
• Examples to compound statements

3
Question If you are under 4 feet tall and you
are younger than 16 years old then you cannot
ride the roller coaster. Translate above
sentence from English to Symbols. Answer p You
can ride the roller coaster qYou are under 4
feet tall rYou are younger than 16 years old
(q ? r ) ? p

• Translating from English to Symbols
• p but q means p and q
• Ex It is not hot but it is sunny.
• (We understand it is not hot and it is sunny)
• Take pIt is hot and qit is sunny
• So our sentence is p ? q
• Neither p nor q means p and q
• Ex It is neither hot nor sunny.
• (We understand it is not hot and it is not
  sunny)
• So our sentence is p ? q

4
Truth Values
Conjunction The conjunction of the statements p
and q is true only when both p and q are true.
Disjunction The disjunction of the statements p
and q is false only when both p and q are false
Conditionals Given two statements p and q, the
new statemant If p then q is called a
conditional statement. p hypothesis q
conclusion p ? q is false only when p true and
q falseWe read p ? q as If p then q or p
implies q Other terminology used for p ? q p
is sucient for q, q whenever p, q follows
from p, q is necessary for p
Biconditionals Given two statements p and q, the
conjunction (p ? q) ?(q ? p) of the conditional
propositions p ? q and q ? p is called
biconditional. We read this as p if and only if
q or p is necessary and sufficient for q
or if p then q, and conversely We denote it by
p ? q. Definition If the biconditional
statement is true, then p and q is said to be
equivalent. In this case, we write pq.
5
Example 1 Evaluate the truth value of p ? q
? (p ? q). We will use truth table.
6
Example 2 Let A and B be two satements.
Construct the truth table for (A ? B) ? (A ?
B).
7

• Definition
• A statement form is an expression made up of
  statement variables (like p, q and r) and logical
  connectives (such as , ?, ?)
• A tautology (denoted by t) is a statement form
  that is always true regardless of the truth
  values of the individual statements.
• A contradiction (denoted by c) is a statement
  form whose negation is a tautology(i.e. a
  statement form which is always false regardless
  of the truth values of the individual
  statements).
• Examples of tautologies
• p ? p
• p ? p
• p ? (p ? p)
• p ? (p ? q)
• (p ? q) ? p
• (p ? (p ? q)) ? q
• ((p ? q) ? q) ? p
• Exercise Draw truth tables of above statement
  forms.
• An example to contradiction p ? p
• NOTE A statement form is a tautology if and only
  if its negation is a contradiction.

8
Question Show that p ?(p ?q )?q is a
tautology .
p ?(p ?q )?q is always true. So, it is a
tautology.
9
Logical Equivalence

• Definition The statements p and q are called
  logically equivalent if they have identical truth
  values, denoted by p q.
• One way to determine if two propositions are
  logically equivalent is to use truth table.
• Example Show that (p ? q) p ? q.

10
Logical Equivalence

• Definition The statements p and q are called
  logically equivalent if they have identical truth
  values, denoted by p q.
• One way to determine if two propositions are
  logically equivalent is to use truth table.
• Example Show that (p ? q) p ? q.

11

• The truth table can also be used to show that two
  propositions are not equivalent.
• So, (p ? q) is NOT equivalent to p ? q.
• De Morgans Laws
• The negation of an and statement is logically
  equivalent to the or stement in which each
  component is negated.
• The negation of an or statement is logically
  equivalent to the and stement in which each
  component is negated.
• So, by De Morgans law, (p ? q) p ?q and
  (p ? q) p ? q

12
A table of Logical Equivalences

• Let T be a tautology and F be a contradiction.
  Let p, q and r be any primitive statements. Then
  the following logical equivalences hold.