phrasebook

# phrasebook - the phrase book basic words and symbols of...

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the phrase book basic words and symbols of higher mathematics c ± University of London, 2003.

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1 introduction The language of higher mathematics has words and symbols. The most basic ones are described in this document; they are very general, and will appear in every course you take. The ﬁrst year lecturers will introduce them within the context of their courses, and will expect you to use them when you communicate mathematical information, orally or in writing. Learning the basic language of higher mathematics and being able to use it with precision and ﬂuency is one of the main objectives of your ﬁrst year at University. Your achievements in this area will be monitored by speciﬁc parts of the assessment. 2 logic If P and Q are mathematical statements, we write P Q to mean that P implies Q or Q follows from P ; we write P Q to mean that P is implied by Q ”. For example x = 5 x 2 = 25 x 2 = 25 x = - 5 x 2 = 25 x = ± 5 . We use the double-headed arrow P ⇐⇒ Q to mean that P implies and is implied by Q . In this case P and Q are equivalent statements —they are both true or both false. The expression “implies and is implied by” is often replaced by the awkward “if and only if” : thus the statement x 2 = 25 ⇐⇒ x = ± 5 may be read out loud as x 2 equals 25 if and only if x equals plus or minus 5 ”, and may also be written as x 2 = 25 iff x = ± 5 . 1
3 sets A set is any collection of distinct objects. The members of a set are called elements , and a set is determined by its elements. In some cases, a set can be deﬁned by listing its elements, separated by commas, between curly brackets: for example { 1 , 2 , 3 } denotes the set whose elements are 1, 2 and 3. Two sets are equal if they have the same elements: for example { 1 , 2 , 3 } = { 3 , 2 , 1 } = { 2 , 1 , 3 , 1 , 3 } . Note that the order in which the elements of a set are listed is irrelevant and repeti- tion is ignored. A set may be ﬁnite or inﬁnite. The number of elements of a set is called its cardinality . Thus the cardinality of { 3 , 4 } is 2. The set {} with no elements is called the empty set , often denoted by the symbol /0. Its cardinality is zero. The empty set is distinct from “nothing” —it is more like an empty container. For the same reason, 3 is distinct from { 3 } , the former being a number, the latter a set having a number as its only element. To indicate that

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phrasebook - the phrase book basic words and symbols of...

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