Difference between revisions of "Math 360, Fall 2021, Assignment 1"
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==Carefully state the following theorems (you do not need to prove them):== |
==Carefully state the following theorems (you do not need to prove them):== |
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+ | # Russell's paradox (this is not really a theorem, but it is an important fact). |
# Basic counting principle (relating the size of $A\times B$ to the sizes of $A$ and $B$). |
# Basic counting principle (relating the size of $A\times B$ to the sizes of $A$ and $B$). |
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# Prove that for any set $S$, it must be the case that $\emptyset\subseteq S$. ''(Hint: begin with "Suppose not. Then by the definition of set containment, there must be some member of $\emptyset$ which is not a member of $S$. But...".)'' |
# Prove that for any set $S$, it must be the case that $\emptyset\subseteq S$. ''(Hint: begin with "Suppose not. Then by the definition of set containment, there must be some member of $\emptyset$ which is not a member of $S$. But...".)'' |
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# Now suppose $S$ is any set with $S\subseteq\emptyset$. Prove that $S=\emptyset$. ''(Hint: use the previous result together with the definition of set equality.)'' |
# Now suppose $S$ is any set with $S\subseteq\emptyset$. Prove that $S=\emptyset$. ''(Hint: use the previous result together with the definition of set equality.)'' |
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| − | # The previous two exercises show that $\emptyset$ is the "smallest of all sets." Is there a "largest of all sets?" (For more information on this question as well as on |
+ | # The previous two exercises show that $\emptyset$ is the "smallest of all sets." Is there a "largest of all sets?" (For more information on this question as well as on Russell's Paradox and its resolutions, see [https://en.wikipedia.org/wiki/Universal_set Wikipedia: Universal Set].) |
<center><big>'''--------------------End of assignment--------------------'''</big></center> |
<center><big>'''--------------------End of assignment--------------------'''</big></center> |
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Revision as of 09:54, 10 September 2021
By one of those caprices of the mind, which we are perhaps most subject to in early youth, I at once gave up my former occupations; set down natural history and all its progeny as a deformed and abortive creation; and entertained the greatest disdain for a would-be science, which could never even step within the threshold of real knowledge. In this mood of mind I betook myself to the mathematics, and the branches of study appertaining to that science, as being built upon secure foundations, and so, worthy of my consideration.
- - Mary Shelley, Frankenstein
Read:
- Section 0.
Carefully define the following terms, then give one example and one non-example of each:
- Containment (of sets).
- Equality (of sets).
- Property.
- Ordered pair.
- Cartesian product (of two sets).
Carefully state the following theorems (you do not need to prove them):
- Russell's paradox (this is not really a theorem, but it is an important fact).
- Basic counting principle (relating the size of $A\times B$ to the sizes of $A$ and $B$).
Solve the following problems:
- Section 0, problems 1, 2, 3, 4, 5, 6, 7, 8, and 11.
- Prove that for any set $S$, it must be the case that $\emptyset\subseteq S$. (Hint: begin with "Suppose not. Then by the definition of set containment, there must be some member of $\emptyset$ which is not a member of $S$. But...".)
- Now suppose $S$ is any set with $S\subseteq\emptyset$. Prove that $S=\emptyset$. (Hint: use the previous result together with the definition of set equality.)
- The previous two exercises show that $\emptyset$ is the "smallest of all sets." Is there a "largest of all sets?" (For more information on this question as well as on Russell's Paradox and its resolutions, see Wikipedia: Universal Set.)
