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04_Transcript - 1 Existential Quantiers The Construction...

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1 Existential Quantifiers: The Construction Method 1.1 Welcome Welcome to the fourth module in our course on proofs,“Existential Quantifiers: The Construction Method”. Before going on, please do the assigned readings. 1.2 Introduction Not all mathematical statements are obviously in the form “If A , then B ”. You will encounter statements of the form there is , there are , there exists or for all , for each , for every , for any . The first three are all examples of the existential quantifier there is and the final four are all examples of the universal quantifier for all . The word existence is used to make it clear that we are looking for or looking at a particular mathematical object. The word universal is used to make it clear that we are looking for or looking at a set of objects all of which share some desired behaviour. This module deals with existential quantifiers. The next module deals with universal quantifiers. 1.3 Basic Structure and Examples All statements which use existential quantifiers share a basic structure, though some elements of the structure may be absent, implicit or appear in a different order. There is an “object” in the “universe of discourse” with a “certain property” such that “something happens”. It is very important to be able to identify the object, the universe of discourse, the property and the something that happens. The expression “such that” (or equivalent words like “for which”) usually precede the something that happens. The universe of discourse is the set from which the “object” is taken. More formally, it is the domain of the object. Here are some examples. 1. There exists a real number x so that f ( x ) = 0. Object: x Universe of discourse: R Certain property: None specified Something happens: f ( x ) = 0 This is a good point to illustrate the universe of discourse. Suppose in this example we are interested in the specific function f ( x ) = x 2 - 2. Then the statement There exists a real number x such that x 2 - 2 = 0. is true since we can find an x , 2, so that x 2 - 2 = 0. But if we change the universe of discourse to integers, There exists an integer x such that x 2 - 2 = 0. 1
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the statement is false because neither of the two roots, 2 or - 2 are integers. So changing the universe of discourse can change the truth value of the statement. In practice, the universe of discourse is often not explicitly stated and is inferred from context. 2. There exists an angle θ such that sin( θ ) = 1. Object: angle θ Universe of discourse: R , inferred from context Certain property: None specified Something happens: sin( θ ) = 1 In this example, no property is specified and the universe of discourse is implicit. Note also that there can be many objects, that is many angles θ , which satisfy the statement.
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