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Past exam of the year 1 of the computer science course of the University of Oxford 2013 Discrete mathematics, probability and continuous mathematics 3 e i Solution by
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The binary relation is reflexive and transitive but is not a symmetric relation. For example, while . ThereforeThere are consequently no equivalence classes for this binary relation in the requested sense.
Past exam of the year 1 of the computer science course of the University of Oxford 2013 Discrete mathematics, probability and continuous mathematics 2 b v Solution by
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Past exam of the year 1 of the computer science course of the University of Oxford 2013 Discrete mathematics, probability and continuous mathematics 2 b iv Solution by
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The exponential function eventually grows faster than the fixed polynomial. A direct justification uses the ratio :For all sufficiently large , this ratio exceeds , so grows without bound. The proposed Big O notation bound is therefore impossible:
Past exam of the year 1 of the computer science course of the University of Oxford 2013 Discrete mathematics, probability and continuous mathematics 2 b iii Solution by
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Divide by the proposed comparison function :The ratio is unbounded, so no constant in the definition of Big O notation can work:
Past exam of the year 1 of the computer science course of the University of Oxford 2013 Discrete mathematics, probability and continuous mathematics 2 b ii Solution by
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The logarithm laws giveFor , both and . The expression is therefore at most . ConsequentlyThe Big O notation concerns eventual behaviour; the zero value of at is irrelevant.
Past exam of the year 1 of the computer science course of the University of Oxford 2013 Discrete mathematics, probability and continuous mathematics 2 b i Solution by
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Past exam of the year 1 of the computer science course of the University of Oxford 2013 Discrete mathematics, probability and continuous mathematics 1 f v Solution by
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The proposed inverse function cancels on both sides:The uniqueness of an inverse function therefore givesThe operation applied last must be undone first.
Past exam of the year 1 of the computer science course of the University of Oxford 2013 Discrete mathematics, probability and continuous mathematics 1 f iv Solution by
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For the bijections and used above, sends to . Its inverse function sends to , whereas sends to . Hence
Past exam of the year 1 of the computer science course of the University of Oxford 2013 Discrete mathematics, probability and continuous mathematics 1 f iii Solution by
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Use the associativity of function composition to cancel adjacent inverse functions in the right order:Therefore
Past exam of the year 1 of the computer science course of the University of Oxford 2013 Discrete mathematics, probability and continuous mathematics 1 f ii Solution by
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The inverse factors are generally in the wrong order. For a counterexample, take the bijections and , with inverse functions and . ThenThusFor particular bijections that commute the equality can hold; the claim fails as a universal assertion.
Past exam of the year 1 of the computer science course of the University of Oxford 2013 Discrete mathematics, probability and continuous mathematics 1 f i Solution by
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The function composition of two bijections is a bijection. Its inverse function is two-sided, soThis is exactly the defining cancellation property of an inverse function, with the whole function composition regarded as one function.
Past exam of the year 1 of the computer science course of the University of Oxford 2013 Discrete mathematics, probability and continuous mathematics 1 d iv Solution by
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Past exam of the year 1 of the computer science course of the University of Oxford 2013 Discrete mathematics, probability and continuous mathematics 1 d iii Solution by
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Past exam of the year 1 of the computer science course of the University of Oxford 2013 Discrete mathematics, probability and continuous mathematics 1 d ii Solution by
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Past exam of the year 1 of the computer science course of the University of Oxford 2013 Discrete mathematics, probability and continuous mathematics 1 d i Solution by
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The image of a function is the set of values actually attained as its argument ranges over its domain of a function:It is a subset of the codomain , and need not equal .
Past exam of the year 1 of the computer science course of the University of Oxford 2013 Discrete mathematics, probability and continuous mathematics 1 c iii Solution by
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Use the symmetric difference formula from part (b), followed by the set difference identity from part (c)(ii):Taking both complements swaps the two ways to belong to exactly one set, leaving their union unchanged:
Past exam of the year 1 of the computer science course of the University of Oxford 2013 Discrete mathematics, probability and continuous mathematics 1 c ii Solution by
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Write the set difference as an intersection and use the fact that taking the complement of a set twice returns that set. The commutativity of intersection gives the reversible calculation
Past exam of the year 1 of the computer science course of the University of Oxford 2013 Discrete mathematics, probability and continuous mathematics 1 c i Solution by
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An algebraic way to prove the set identity is to show that is a complement of a set for . By the distributive property,andA complement of a set is unique: if both complement , then , and similarly . Consequently
Past exam of the year 1 of the computer science course of the University of Oxford 2013 Discrete mathematics, probability and continuous mathematics 1 a ii Solution by
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Apply the distributive property to the intersection with :In the first line the repeated factor changes nothing. Rewriting in terms of set differences yields
Past exam of the year 1 of the computer science course of the University of Oxford 2013 Discrete mathematics, probability and continuous mathematics 1 a i Solution by
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Write each set difference as an intersection with a complement of a set. Applying De Morgan's laws and then the distributive property givesThe complements turn the inner union into an intersection and vice versa. Thus
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
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This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
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