midterm1sols

midterm1sols - Introduction to Advanced Mathematics Midterm...

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Introduction to Advanced Mathematics — Midterm I September 29, 2011 Where appropriate, write your answers in the space provided after each question. Please make an effort to write neatly. Your Name: KEY 1
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(1) — Basic notation. Express in correct mathematical notation the set of all integers which are perfect squares, that is, squares of other integers. { n Z |∃ m Z , n = m 2 } or { m 2 | m Z } Popular incorrect answers: { n Z | n = m 2 } (what is m ?); { n Z | n 2 } ( n 2 is a number, not a statement that can be true or false); {∃ n Z ... } , {∀ n Z ... } (that’s not how we write sets); { n,m Z | n = m 2 } (ambiguous notation—is this a set of n Z , or of m Z ?); { n Z |∀ m Z , n = m 2 } (this would be the set of numbers which are simultaneously squares of all integers, there are no such beasts); ‘ { n | n S } , where S is the set of perfect squares’ (you can’t use S to define S ), etc. Explain the difference between { 1 , 2 , 3 } and (1 , 2 , 3) . { 1 , 2 , 3 } denotes the set consisting of the numbers 1 , 2 , 3 , while (1 , 2 , 3) denotes an ordered triple of numbers, an element of a Cartesian product. Explain the difference between and {∅} . denotes the empty set, with no elements, while {∅} denotes the set with one element, that element being the empty set. The first set is empty, the second set is not empty. Explain the difference between the statements ( x R )( y R ) x < y and ( y R )( x R ) x < y The first statement asserts that for every real number there exists a larger real number; this is a true statement. The second asserts that there exists a real number which is larger than every real number; this is a false statement. Remark: See § 1 of the notes if anything about all this is unclear. 2
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(2) — Parsing statements involving quantifiers. Denote by x real numbers and by N natural numbers. Are x > 0 N : n > N = ⇒ | s n | < x and x > 0 N : n > N = ⇒ | s n | < 3 x equivalent statements, in the sense that they express the same condition on any indexed set { s n } n N of real numbers? (Check Y if they are, N if they are not.) Y
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midterm1sols - Introduction to Advanced Mathematics Midterm...

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