3. Discrete vs. Continuous Mathematics
Continuous Mathematics
It considers objects that vary continuously;
Example:
– Analog wristwatch (separate hour, minute, and second hands).
– 1:25 pm to 1:27pm
– Real-number system --- core of continuous mathematics;
Discrete Mathematics
It considers objects that vary in a discrete way.
Example:
– Digital wristwatch.
– Integers --- core of discrete mathematics
4. Problems Covered
• Cover non-continuous domain. For example,
• Is it possible to visit 3 islands in a river with 6
bridges without crossing any bridge more than
once?
• Or, what is the smallest number of telephone
lines needed to connect 200 cities?
5. Logic
• The rules of logic give precise meaning to
mathematical statements.
• These rules are used to distinguish between
valid and invalid mathematical arguments.
6. Statement
• A statement is declarative sentence that is
either true or false, but not the both.
• A statement is also referred as a proposition.
7. Propositional Logic/ Calculus
• The area of logic that deals with propositions is called the propositional
calculus or propositional Logic.
• Example of proposition logic:
– Washington, D.C., is the capital of the United States of America.
– Toronto is the capital of Canada.
– 1 + 1 = 2.
– 2 + 2 = 3.
• Example of not proposition logic
– What time is it?
– Read this carefully.
– x + 1 = 2.
– x + y = z.
• Sentences 1 and 2 are not propositions because they are not declarative
sentences. Sentences 3 and 4 are not propositions because they are
neither true nor false
8. Propositional Variables
• Variables that represent propositions, just as
letters are used to denote numerical variables.
• If a proposition is true, we say that it has a
truth value of “true” and denoted by T.
• If a proposition is false, we say that it has a
truth value of “false” and denoted by F.
9. Compound Proposition
• Many mathematical statements are
constructed by combining (logical operators)
one or more propositions.
• Examples
• “3+2=5” and “Kurukshetra is a city”
• “the grass is green” or “it is raining”
• “Discrete mathematics is a subject not difficult
to me”
10. Compound Propositions
• Negation (not) p
• Conjunction (and) p q
• Disjunction (or) p q
• Exclusive or p q
• Implication p q
• Biconditional p q
11. Negation
• Let p be a proposition. The negation of p,
denoted by￢p (also denoted by p), is the
statement
• “It is not the case that p.”
• The proposition ￢p is read “not p.” The truth
value of the negation of p, ￢p, is the opposite
• of the truth value of p.
12. Example of Negation
• Find the negation of the proposition
“Michael’s PC runs Linux” and express this in
simple English.
• Solution: The negation is “It is not the case
that Michael’s PC runs Linux.”
This negation can be more simply expressed as
“Michael’s PC does not run Linux.”
13. Question
• Find the negation of the proposition
“Vandana’s smartphone has at least 32GB of
memory” and express this in simple English.
14. Conjunction (And)
• Let p and q be propositions. The conjunction of
p and q, denoted by p ∧ q, is the proposition
“p and q.” The conjunction p ∧ q is true when
both p and q are true and is false otherwise.
15. Disjunction
• Let p and q be propositions. The disjunction of
p and q, denoted by p ∨ q, is the proposition
“p or q.” The disjunction p ∨ q is false when
both p and q are false and is true otherwise.
