1. In the first chapter of our textbook, Section 1.1 \"The origins of Mathematic
ID: 2246924 • Letter: 1
Question
1. In the first chapter of our textbook, Section 1.1 "The origins of Mathematical Logic", the author states
that Logic must be formalized because reasoning expressed in informal natural language can be flawed.
As a proof, he cites an example of a flawed syllogism given in 1978 by Smullyan:
Premise: Some cars rattle.
Premise: My car is some car.
Conclusion: My car rattles.
What is wrong with that syllogism?
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2. The Lecture Notes states "Eventually it (Mathematical Logic) became also part of the foundations of
Computer Science, serving both a theoretical and a practical purpose." Please explain:
(a) The theoretical importance of Mathematical Logic
(b) The practical importance of Mathematical Logic
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3. Let P, Q and R be sets, P R, Q R. Are the following expressions true or false? Explain why.
(a) P = (P Q) – P
(b) P Q = (P – Q) Q
(c) (P Q)C = PC Q
C
(d) P R = P
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4. Let P, Q and R be the sets
P = { n | n , 2 n 10 }
Q = { n | n , 5 < n 20 }
R = { n | n , 1 n < 10 }
Find:
(a) P (Q R)
(b) QC (P – R)
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5. Let S = {1, 2, 3, 4}. List the members of the powerset of S.
(The powerset of a set S, denoted 2S
, is defined as the set of all subsets of S, including and S itself)
Explanation / Answer
1. In the first chapter of our textbook, Section 1.1 "The origins of Mathematical Logic", the author states
that Logic must be formalized because reasoning expressed in informal natural language can be flawed.
As a proof, he cites an example of a flawed syllogism given in 1978 by Smullyan:
Premise: Some cars rattle.
Premise: My car is some car.
Conclusion: My car rattles.
What is wrong with that syllogism?
Answer :
We know that some car Rattles. That means all the cars in the universe do not rattle only few out of them rattle.
My car is some car. : This cannot be true because all the car in the universe belongs to larger set
Hence the conclusion cannot be true
2. The Lecture Notes states "Eventually it (Mathematical Logic) became also part of the foundations of
Computer Science, serving both a theoretical and a practical purpose." Please explain:
(a) The theoretical importance of Mathematical Logic:
mathematical logic included set theory, graph theory proofs etc . It has been ongoing from late 19th centry and it has proved that set theory has formalised ordinary mathematics in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory.
This is how theoretical importance of Mathematical Logic has gained its demand.
(b) The practical importance of Mathematical Logic
Practical importance : Logic is used everywhere. We use mathametical logic in Digital circuits like Counters, registers. Its been implemented practically. Its far more useful and used in everydays life.
We used Fuzzy logic which comes useful in Search. We might have seen Washingmachine uses such logic so its used everywhere.
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3. Let P, Q and R be sets, P ? R, Q ? R. Are the following expressions true or false? Explain why.
(a) P = (P ? Q) – P
P is a set inside R , Let say : R = {1,2,3,4,5}
P = {1,2}
Q= {2,4}
P U Q = {1,2,4}
{1,2,4} - P = {4}
So, it cannot be TRUE
(b) P ? Q = (P – Q) ? Q
P is a set inside R , Let say : R = {1,2,3,4,5}
P = {1,2}
Q= {2,4}
P - Q = {1}
{1} U Q = {1,2,4}
P U Q = {1,2,4}
So, its True
(c) (P ? Q)c= Pc ? Qc
( P U Q )c = {1,2,4} = {3,5 ..}
Pc = {3,4,5...}
Qc = {1,3,5...}
Pc ? Qc = {1,3,4,5.....}
So it cannot be TRUE
(d) P ? R = P
This will always be True because P is subset of R
4. Let P, Q and R be the sets
P = { n | n ? ?, 2 ? n ? 10 }
P = {2,3,4,5,6,7,8,9,10}
Q = { n | n ? ?, 5 < n ? 20 }
Q = {6,7,8,9,10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}
R = { n | n ? ?, 1 ? n < 10 }
R = {1, 2,3,4,5,6,7,8,9}
Find:
(a) P ? (Q ? R)
Q U R = {1, 2,3,4,5,6,7,8,9,10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}
P ? (Q ? R) = {2,3,4,5,6,7,8,9,10}
(b) QC ? (P – R)
(P – R) = {10}
QC = {1,2,3,4,5 , 21 .. ... ETC}
QC ? (P – R) = {?}
5. Let S = {1, 2, 3, 4}. List the members of the powerset of S.
(The powerset of a set S, denoted 2S is defined as the set of all subsets of S, including ? and S itself)
Power Set = { {?} , {1} , {2} , {3} , {4} , {1,2} , {1,3} , {1,4} , {2,3} , {2,4} , {3,4} , {1,2,3} , {1,2,4} , {1,3,4} , {2,3,4}, {1,2,3,4} }
Thanks, let me know if there is any concern.
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