Express the negation of each of these statements in terms of...

Express the negation of each of these statements in terms of quantifiers without using the negation symbol. a) $\forall x(-2<x<3)$ b) $\forall x(0 \leq x<5)$ c) $\exists x(-4 \leq x \leq 1)$ d) $\exists x(-5<x<-1)$

Transcript

00:01 In this problem, we need to express the negation of each of the given statements.

00:06 Now, the first statement is, for all x minus 2 less than x, less than 3.

00:18 So, we need to write the negation of this statement.

00:23 Now, the negation of a statement of the form for all x, then we have something.

00:29 We will have there exists and x says that naught of whatever is given.

00:36 So this will be the negation.

00:39 So we need to make sure that we do not have the negation symbol.

00:44 So let us simplify this.

00:45 So this will be there exists and x says that naught of minus 2 less than x, less than 3 means minus 2 is less than x and x is less than 3.

00:57 So we need to negate that.

01:01 So by d morgan's laws, the negation of p and q is not p or not q.

01:09 So we need to write the negation of minus 2 less than x, so that will be minus 2 greater than or equal to x.

01:16 Then instead of and we have or, then we have not of x less than 3.

01:21 So that will be x is greater than or equal to 3.

01:24 So this is our negation.

01:25 There exists x minus 2 greater, greater, equal to x or x is greater equal to 3.

01:31 Then for the next statement, the next statement is for all x, 0 is less than or equal to x is less than 5.

01:43 So the negation will be there exists an x, so is that not 0 less than or equal to x less than 5.

01:58 So this is there exists, there exist and x is that knot of 0 is less than or equal to x and x is less than 5.

02:11 So we use d.

02:13 Morgan's law and the negation over here will be of the form not p or not q...

Question

Express the negation of each of these statements in terms of quantifiers without using the negation symbol. a) $\forall x(-2<x<3)$ b) $\forall x(0 \leq x<5)$ c) $\exists x(-4 \leq x \leq 1)$ d) $\exists x(-5<x<-1)$

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