The notation ∃x P(x) denotes the proposition "There exists a...

Key Concepts

Unique Existence Quantifier

This concept involves the use of a quantifier that asserts there is exactly one element in a given domain that satisfies a particular property. It combines both existence (at least one element) and uniqueness (no more than one element), and is usually denoted by the symbol ?! or phrasing like 'there exists a unique x'.

Domain of Discourse

The domain of discourse is the set of elements over which the variables in a logical proposition range. In problems like this, specifying that the domain consists of all integers is essential, since the truth of statements regarding existence and uniqueness can change when the underlying set of elements (for example, integers, real numbers, etc.) is altered.

Solving Equations and Inequalities in the Context of Unique Solutions

Evaluating the truth of propositions involving equations or inequalities requires solving these expressions within the specified domain. In the context of unique existence, one must verify not only that a solution exists, but that it is the only solution. For example, determining whether an equation like x + 3 = 2x has exactly one solution in the set of integers involves both finding the solution and checking its uniqueness.

Distinguishing Between Existence and Uniqueness

It is important to differentiate between merely having a solution (existence) and having precisely one solution (uniqueness). A statement may be true in terms of existence but false when the uniqueness condition is imposed. This distinction is critical when interpreting and analyzing logical statements that use a unique existence quantifier.

Transcript

00:01 Alright, in this problem we are going to be exploring this idea of an existential quantifier that happens to be unique.

00:08 So in particular what we're going to be looking at is this backward e, which means there exists, and we're going to attach this exclamation point, which means there exists uniquely.

00:21 And in particular we're going to be looking at if there exists uniquely in x, such that the statement p of x is true.

00:30 And also note that along with with this we're going to be restricting where our x is coming from, where does x live.

00:39 And so note that x is going to be an element of the set of integers.

00:45 And recall that the set of integers is the set of all positive and negative whole numbers.

00:51 So dot dot dot comma negative two negative one zero one two dot dot dot.

00:59 So we're looking at all of those.

01:02 But the key to this problem is really understanding this notation.

01:07 Right, what does this notation mean? there exists uniquely.

01:13 So again the way that we read that is that there exists uniquely.

01:24 That means first and foremost there needs to be at least one, there needs to be something that meets the condition stated in our in our statement p of x.

01:34 But the exclamation point again adds this uniqueness quality to it, which means that there can be only one.

01:41 So another way that we can read this is we can say there is one, because that has to exist, but we can add and only, right, because it has to be unique.

02:00 It has to be the only one.

02:01 So another way that we could read this is there is one and only one x such that our statement p of x is true.

02:11 Alright so now we're going to go ahead and take a look at four different statements.

02:17 And so let's go ahead we're going to add this existential quantifier, this unique existential quantifier to our statement.

02:24 So there exists uniquely an x such that our statement p one of x is true.

02:30 And we're going to define p one of x to be that x is greater than one.

02:38 Okay so if we start thinking about x is being greater than one, then again x comes from the set of integers.

02:46 The first thing that we need to do is check.

02:49 Is there at least one? is there is there one integer that's greater than one? well yes, right, there is.

02:57 I can say let's pick three for example.

03:01 Right if x is equal to three, well then that means that it meets our condition.

03:07 Three is greater than one.

03:08 So there is at least one.

03:10 There's one integer at least that that makes my statement p one of x true.

03:16 But is it the only one? okay so we've met our existence condition.

03:21 Now we're going to check uniqueness.

03:22 Is it the only one? well no, you know it's not.

03:26 In fact there's an infinite number of integers that are larger than one.

03:30 One instance is the number ten.

03:34 Right x if x is equal to ten, ten is greater than one.

03:38 And so here we have something that's not unique.

03:42 It's not the only one.

03:44 So our statement then there exists uniquely an x such that p one of x is true is false.

03:51 This is false because it's not unique.

03:56 There's one that exists but it's not unique.

03:59 Alright let's check our next statement.

04:02 Okay so there exists again uniquely an x such that p2 of x is true.

04:09 And we're going to define our p2 of x to be that x squared x squared is equal to one.

04:20 So again the first thing that we want to do is check to see if it exists.

04:24 Is there an x that makes this statement true? well is there an integer that i can times by itself to give me one? well yes there is.

04:34 If i let x be equal to one, right, one is an integer.

04:39 And when i square it, one squared gives me one.

04:42 So it exists.

04:44 Right so we have that that it does exist.

04:48 Now we again we have to check our second condition here.

04:50 Is it unique? is it the only one? well on the outset it seems like it maybe it is because we have you know two squared is is going to be four.

05:02 Three squared is nine...

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