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In our everyday lives we use we use language to develop ideas and to communicate them...
In our everyday lives we use we use language to develop ideas and to communicate them to other people. In this unit we examine ways in which language is adapted to express mathematical ideas.
By the end of this unit you should be able to:
- Section 1: Sets
- use set notation;
- determine whether two given sets are equal and whether one given set is a subset of another;
- find the union, intersection and difference of two given sets.
- Section 2: Functions
- determine the image of a given function;
- determine whether a given function is one-one and/or onto;
- find the inverse of a given one-one function;
- find the composite of two given functions.
- Section 3: The language of proof
- understand what is asserted by various types of mathematical statements, in particular implications and equivalences;
- produce simple proofs of various types, including direct proof, proof by induction, proof by contradiction and proof by contraposition;
- read and understand the logic of more complex proofs;
- disprove a simple false implication by providing a counter-example.
- Section 4: Two identities
- understand and use the Binomial Theorem;
- understand and use the Geometric Series Identity;
- understand and use the Polynomial Factorisation Theorem.
When we try to use ordinary language to explore mathematics, the words involved may not have a precise meaning, or may have more than one meaning. Many words have meanings that evolve as people adapt their understanding of them to accord with new experiences and new ideas. At any given time, one person's interpretation of language may differ from another person's interpretation, and this can lead to misunderstandings and confusion.
In mathematics we try to avoid these difficulties by expressing our thoughts in terms of well-defined mathematical objects. These objects can be anything from numbers and geometrical shapes to more complicated objects, usually constructed from numbers, points and functions. We discuss these objects using precise language which should be interpreted in the same way by everyone. In this unit we introduce the basic mathematical language needed to express a range of mathematical concepts.
Please note that this unit is presented through a series of downloadable PDF files.
This unit is an adapted extract from the Open University course