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Elementary Logic
Æ p is called the
converse of p Æ q, and ¬q Æ ¬p is the
contrapositive of p Æ q
• Example:
propositions
(^2) = x, then x = 0 or x = 1
above
propositions
Example :
go shopping.
P: I go to Harry’s Q: I go to the country R: I will go shopping If......P......or.....Q.....then....not.....R
(P∨Q)→￢R
• The proposition q
- Give the converse of the following
- If I am smart, then I am rich
- If x
- If 2 + 2 = 4, then 2 + 4 = 8
- Give the contrapositives for the propositions
Breaking assertions into component
- look for the logical operators!
If I go to Harry’s or go to the country I will not
Elementary Logic
p, q, r and logical connectives
are clouds in the sky
Convert the following into predicate
logic sentences
can
can do a trick
Let p, q, r be the following propositions p = “it is raining” q = “the sun is shining” r = “there are clouds in the sky” Translate the following into logical notation, using
It is raining and the sun is shining If it is raining, then there are clouds in the sky If it is not raining, then the sun is not shining and there
The sun is shining if and only if it is not raining If there are no clouds in the sky, then the sun is shining
• Shamu can do every trick
• Shamu can do any trick
• Shamu cannot do every trick
• If any whale can do a trick, Shamu can
• If every whale can do a trick, Shamu
• If any whale can do a trick, any whale
Proof by Cases
exhaustive —i.e., that include all the
possibilities
• Example : Prove that n^2 -2 is not
dividable by 5 for any positive integer
Proof by Contradiction
assuming it does not. If this leads to
hypotheses), then we have reached a
contradiction.
• Example : Prove that there are
infinitely many primes.
• Example : Prove that the sum of a
rational number and an irrational
number is always irrational.
• Consider several cases that are
- Case 1: n=5k
- Case 2: n=5k+
- Case 3: n=5k+
- Case 4: n=5k+
- Case 5: n=5k+
• We show that a conclusion holds by
‘nonsense’ contrary to reality (or
Direct Proof
simple combination of existing theorems
with/without some mathematical
manipulations
- H 1 ∧ H 2 ∧ … ∧ H (^) n ⇒ C
- ¬C ⇒ ¬(H 1 ∧ H 2 ∧ … ∧ H (^) n)
Direct or Indirect Proof?
• Example : Let m, n ∈ Ν. Prove that if
m+n >= 73 then m >= 37 or n >= 37
• Show that a given statement is true by
• Proof of the contrapositive (indirect proof)
- Example : Convert each of the following
arguments into logical notation using
the suggested variables. Then provide a
formal proof.
the electric bill, then I will run out of
power will be turned off. Therefore, if I do not run out of money and the power is still on, then my computations are incorrect”. (c, b, r, p)
∧b) Æ r and ¬bÆ ¬p, then (¬r∧p)Æ ¬c
- “if my computations are correct and I pay
money. If I do not pay the electric bill, the
- Let
- c := “my computations are correct”
- b := “I pay the electric bill”
- r := “I run out of money”
- p := “the power stays on”
- Then theorem is:
Mathematical Induction
of propositions. If
(B)
(I)
p(k) is true and m ≤ k < n,
then all propositions are true.
• Example : for each positive integer n, let
p(n) be “ n! > 2n
claim is true for n ≥ 4.
To give a proof by induction, we verify that
p(n) for n=4, and then show
≤ k and k!>2k, then (k+1)! > 2k+
Heidi C. Perry / Draper Labs.
- When : Wednesday 5/
Topic: What Software Do You Need to Get to
intensive. aircraft, rotorcraft, submersibles, missiles and spacecraft, have been revolutionized because of the integration of
of these systems extensively use information generated by a myriad of sensor subsystems that are highly integrated into
a number of examples of highly reliable, flight critical software applications.
• Let p(m), p(m+1), …, p(n) be a sequence
p(m) is true, and
p(k+1) is true whenever
,” a proposition that we
(I) If 4
Division Leader - Software Engineering
Mars? A Look at Large Scale Software Development Today's aerospace applications are extremely software The designs of all modern flight platforms, e.g.,
advanced microelectronics and complex sensor systems. All
a real-time embedded software architecture. This lecture will explore what it takes to develop large scale software for
