Find a counterexample, if possible, to these universally...

Find a counterexample, if possible, to these universally quantified statements, where the domain for all variables consists of all integers.
a) $\forall x \exists y(x=1 / y)$
b) $\forall x \exists y\left(y_{2}^{2}-x<100\right)$
c) $\forall x \forall y\left(x^{2} \neq y^{3}\right)$

Key Concepts

Properties of Integers

Understanding the properties of integers, such as divisibility, the behavior of inverse operations, and relationships involving powers (like squares and cubes), is crucial when constructing or refuting statements in number theory. These inherent properties provide the foundation for testing the truthfulness of logical assertions involving integers.

Counterexample

A counterexample is a specific instance or element from the domain that demonstrates the falsity of a universally quantified statement. By providing an example that does not satisfy the conditions of the statement, it proves that the statement does not hold in all cases.

Domain of Discourse

The domain of discourse refers to the set of all elements under consideration for a given logical statement. It defines the boundaries within which the quantifiers are evaluated, such as the set of all integers when dealing with arithmetic properties and relations.

Universal Quantification

Universal quantification refers to the assertion that a certain property or relation holds for all elements within a specified domain. It is used to state that no matter which element of the domain you pick, the property must be true or the statement holds for every instance.

Existential Quantification

Existential quantification is the aspect of a statement that claims there is at least one element in the domain for which a given property or relation holds true. It is used to indicate the existence of some element that satisfies the condition, contrasting with the universality of universal quantification.

Transcript

00:02 So for this problem, we're given three universally quantified statements, and we want to see if we can find a counter example to each of these.

00:09 So keep in mind that each of these x and y values are going to be integers.

00:15 So let's start by looking at part a.

00:18 So this statement says that for every x, there exists a y such that x is equal to 1 over y.

00:25 So we can take any value of x, any integer, and it's saying that we can set it equal to 1 divided by some number.

00:32 We can find that number.

00:33 Now the thing that we would have to think about is, can we just fit any old number in for y? can we always find this value for y? and the tricky thing here is that it has a fraction.

00:49 So we have 1 divided by y.

00:51 Y is an integer.

00:53 Is there any integer that we can't divide by? like, is there any number that we could plug in for y that would just render this whole thing useless, that we just would completely mess it up? so remember that we can't divide by 0.

01:09 Right? 1 divided by 0 doesn't make any sense.

01:12 So if we set y to be equal to 0, then we couldn't really find an x.

01:17 There would be no value x that's equal to 1 over 0, because 1 over 0 isn't defined.

01:24 So our counter example here is, say, y equals 0.

01:29 Because there is no value x that's equal to 1 divided by 0.

01:33 For part b, this statement is saying that for every x, there exists a y such that y squared minus x is less than 100.

01:47 So this one, again, we're saying for every x, so any value that we plug in x, we can find a particular value for y such that this is true.

01:57 The thing to keep in mind here is that y squared is always going to be a positive number.

02:05 If we square a number, we can't get an output that's negative.

02:11 So just kind of as a note, y squared is always going to be greater than or equal to 0...

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