14 KiB
| title | TARGET DECK | FILE TAGS | tags | |
|---|---|---|---|---|
| Floors & Ceilings | Obsidian::STEM | algebra algorithm |
|
Overview
The floor of x is the greatest integer less than x. The ceiling of x is the least integer greater than x. These values are denoted \lfloor x \rfloor and \lceil x \rceil respectively.
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Basic
How is the floor of x denoted?
Back: \lfloor x \rfloor
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
What is the floor of x?
Back: The greatest integer less than or equal to x.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
END%%
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Basic
How is the ceiling of x denoted?
Back: \lceil x \rceil
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
END%%
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Basic
What is the ceiling of x?
Back: The least integer greater than or equal to x.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
When does \lfloor x / 2 \rfloor = \lceil x / 2 \rceil?
Back: When x is even.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
When does \lfloor x / 2 \rfloor \neq \lceil x / 2 \rceil?
Back: When x is odd.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
What does \lceil x \rceil - \lfloor x \rfloor equal?
Back: Either 0 or 1.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
What can be said about x if \lceil x \rceil - \lfloor x \rfloor = 0?
Back: x is an integer.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
What can be said about x if \lceil x \rceil - \lfloor x \rfloor = 1?
Back: x is not an integer.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
What values of x satisfy \lfloor x \rfloor = \lceil x \rceil - 1?
Back: Non-integral values.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
What values of x satisfy \lceil x \rceil = \lfloor x \rfloor + 1?
Back: Non-integral values.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
END%%
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Basic
What values of x satisfy \lfloor x \rfloor = \lceil x \rceil + 1?
Back: None.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
END%%
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Basic
What values of x satisfy \lceil x \rceil = \lfloor x \rfloor - 1?
Back: None.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
For what values k is horizontal and vertical shifting of \lfloor x \rfloor the same?
Back: Integral values.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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%%ANKI Basic What C operator corresponds to floor division? Back: None. Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
What C operator corresponds to integer division?
Back: /
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
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Basic
When does C operator / behave like floor division?
Back: When the result is a nonnegative value.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
END%%
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Basic
When does C operator / behave like ceiling division?
Back: When the result is a nonpositive value.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
END%%
%%ANKI Basic What C operator corresponds to ceiling division? Back: None. Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
END%%
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Basic
How does C evaluate 10 / 3?
Back: 3
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
Tags: c17
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Basic
How does C evaluate floor(10.f / 3)?
Back: 3
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
Tags: c17
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Basic
How does C evaluate ceil(10.f / 3)?
Back: 4
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
Tags: c17
END%%
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Basic
How does C evaluate -10 / 3?
Back: -3
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
Tags: c17
END%%
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Basic
How does C evaluate floor(-10.f / 3)?
Back: -4
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
Tags: c17
END%%
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Basic
How does C evaluate ceil(-10.f / 3)?
Back: -3
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
Tags: c17
END%%
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Basic
Given r = \lfloor (p + q) / 2 \rfloor, fair partitioning requires A[r] to be included in which of A[p..r-1] or A[r+1..q]?
Back: A[p..r-1]
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
END%%
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Basic
Given r = \lfloor (p + q) / 2 \rfloor, when is A[p..r] or A[r+1..q] equally sized?
Back: When A[p..q] has even size.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
END%%
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Basic
Given r = \lceil (p + q) / 2 \rceil, fair partitioning requires A[r] to be included in which of A[p..r-1] or A[r+1..q]?
Back: A[r+1..q]
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
END%%
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Basic
If A[p..q] has odd size, what r most fairly allows partitions A[p..r] and A[r+1..q]?
Back: r = (p + q) / 2
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
END%%
%%ANKI
Basic
If A[p..q] has odd size, what r most fairly allows partitions A[p..r-1] and A[r..q]?
Back: r = (p + q) / 2
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
END%%
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Basic
If A[p..q] has odd size, what r ensures A[p..r-1] has same size as A[r+1..q]?
Back: r = (p + q) / 2
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
END%%
%%ANKI
Basic
If A[p..q] has even size, what r most fairly allows partitions A[p..r] and A[r+1..q]?
Back: r = \lfloor (p + q) / 2 \rfloor
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
END%%
%%ANKI
Basic
If A[p..q] has even size, what r most fairly allows partitions A[p..r-1] and A[r..q]?
Back: r = \lceil (p + q) / 2 \rceil
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
END%%
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Basic
Given A[p..q] and r = \lfloor (p + q) / 2 \rfloor, how does the size of A[p..r] compare to A[r+1..q]?
Back: It either has zero or one more members.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
END%%
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Basic
Given A[p..q] and r = \lfloor (p + q) / 2 \rfloor, what is the size of A[p..r] in terms of n = q - p + 1?
Back: \lceil n / 2 \rceil.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, 3rd ed (Cambridge, Mass: MIT Press, 2009).
END%%
%%ANKI
Basic
Given A[p..q] and r = \lfloor (p + q) / 2 \rfloor, what is the size of A[r+1..q] in terms of n = q - p + 1?
Back: \lfloor n / 2 \rfloor.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, 3rd ed (Cambridge, Mass: MIT Press, 2009).
END%%
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Basic
Given A[p..q] and r = \lceil (p + q) / 2 \rceil, how does the size of A[p..r-1] compare to A[r..q]?
Back: It either has zero or one fewer members.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
END%%
%%ANKI
Basic
Given A[p..q] and r = \lceil (p + q) / 2 \rceil, what is the size of A[r..q] in terms of n = q - p + 1?
Back: \lceil n / 2 \rceil.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, 3rd ed (Cambridge, Mass: MIT Press, 2009).
END%%
%%ANKI
Basic
Given A[p..q] and r = \lceil (p + q) / 2 \rceil, what is the size of A[p..r-1] in terms of n = q - p + 1?
Back: \lfloor n / 2 \rfloor.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, 3rd ed (Cambridge, Mass: MIT Press, 2009).
END%%
%%ANKI
Basic
Given A[p..q] and r = \lfloor (p + q) / 2 \rfloor, how does the size of A[p..r-1] compare to A[r..q]?
Back: It either has one or two fewer members.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
END%%
%%ANKI
Basic
Given A[p..q] and r = \lceil (p + q) / 2 \rceil, how does the size of A[p..r] compare to A[r+1..q]?
Back: It either has one or two more members.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
END%%
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Basic
Given A[p..q] and r = \lfloor (p + q) / 2 \rfloor, why is the size of A[p..r] potentially larger than A[r+1..q]?
Back: If A[p..q] has odd size, A[p..r] contains the midpoint.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
END%%
References
- Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
- Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
- Thomas H. Cormen et al., Introduction to Algorithms, 3rd ed (Cambridge, Mass: MIT Press, 2009).