Discrete Mathematics and applications Autumn 2010. Tôi là Moshe Rosenfeld Tôi sinh ra ở Israel Tôi đã học toán học tại Đại học Hebrew ở Jerusalem Tôi đến từ trường Đại học Washington Tôi không biết nói tiếng Việt. What Is Mathematics?. 1. 2. 4. 7. 11. 22. 29. 37. ??. 16.
37 + 9 = 46
Did you see the pattern?
Can you tell which number will be in location number 1000? 2,000,000?
Mathematics is the Study of Numbers, Shapes and Patterns
p = “it is hot today” then p = “it is not hot today”
(p vq vw v u …) is called a disjunction.
(p q w u…) is called a conjunction.
What did they eat and where?
(A v B) (C v D) ( A V E) = T
v BC( A ) v BCE v BD( A ) v BDE = T
The only possible true statement is:
BC( A )
From K. Rosen's book.
Five friends have access to a chat room. Is it possible to determine who is chatting if the following information is known?
1. Either Kevin or Heather or both, are chatting.
2. Either Randy or Vijay, but not both are chatting.
3. If Abby is chatting, so is Randy.
4. Vijay and Kevin are either both chatting or neither are chatting.
5. If Heather is chatting, then so are Abby and Kevin.
A : Abbey is chatting.
H : Heather is chatting
R : Randy is chatting
V : Vijay is chatting
K : Kevin is chatting
We can now build a compound statement based on the given data:
(H or K) and (xor(R,V) and implies(A,R) and (not (xor(K,V)) and
(implies(H, (A and K)) = T
to solve this puzzle?
Can we write a computer program
to solve this puzzle?
for H in [true, false]:
for K in [true, false]:
if ((H or K) and (xor(R,V) and implies(A,R)
and (not (xor(K,V)) and
(implies(H, (A and K)) == T):
print(`Heather is chatting: `,H, `\nAbbey is chatting: `, A,
`\nKevin is chatting: `,K,`\nRandy is chatting: `,R,
`\nVijay is chatting: `,V );
Heather is chatting: false
Abbey is chatting: false
Kevin is chatting: true
Randy is chatting: false
Vijay is chatting: true
Example: p p
2. A compound statement which is always FALSE is called a Contradiction
Example: p p
Show that the implication [p(pq)] q is a tautology.
Ans. Need to show that the compound statement is always TRUE.
p q (p q)p(pq)p(pq) q
Show that: (p q) r and p(qr) are not equivalent.
It is common to refer to variables associated with propositions as Boolean.
It is also a common practice to use the “+” sign for “or”, the “.” for “and” and the
(over-strike) for “not”
For example: (p q) (p r)will be written as: (p + q).(p + q)
Truth tables can be used to check the validity of arguments.
Mathematical proofs are a sequence of propositions: p1 p2 ...p2n | q
Where each p2i is either a premise (given true proposition) or an inference and q is the conclusion. The rules are broken down as shown in the table:
We observed that every truth table can be constructed using only “not”, “and”, and “or” operators.
Theoretically and physically there are more gates.
A fundamental problem in both mathematics and computer science is:
Given a compound proposition is there an efficient way to decide whether it is satisfiable?
A “simpler” looking problem but actually equivalent is:
Is there an efficient way to decide whether a given 3-SAT instance is satisfiable?
The logic gates were the bridge that enabled us to explore the many applications provided by computers.
Walther Bothe invented the first modern electronic AND gate. He received the Nobel prize in physics in 1954.
Further interesting information can be found at: