a) Write the negation of each of the following statements: p: For every real number x, x²x; q: For every real number x, either x1 or x

Question

a) Write the negation of each of the following statements: p: For every real number x, x²>x; q: For every real number x, either x>1 or x<1. b) "Mathematics is fun" - check whether this sentence is a statement.

Accepted Answer

> > [!TLDR] > Negating a universal ('for every') gives an existential ('there exists') with the inner condition negated; a sentence is a statement only if it is objectively true or false, which a subjective claim like 'fun' is not. > > **a) Negation of $p$:** 'For every real number $x$, $x^2>x$.' Negating $\forall x, x^2>x$ gives $\exists x$ such that $x^2\le x$. So: *'There exists a real number $x$ such that $x^2\le x$.'* > > **Negation of $q$:** 'For every real number $x$, either $x>1$ or $x<1$.' Negating $\forall x,(x>1\lor x<1)$ gives $\exists x$ such that $ > eg(x>1)\land > eg(x<1)$, i.e. $x\le1$ and $x\ge1$, i.e. $x=1$. So: *'There exists a real number $x$ such that $x=1$ (i.e., neither $x>1$ nor $x<1$).'* > …

Q.a) Write the negation of each of the following statements: p: For every real number x, x²>x; q: For every real number x, either x>1 or x<1. b) "Mathematics is fun" - check whether this sentence is a statement.

Concept understanding — Negation of a Statement

Q.a) Write the negation of each of the following statements: p: For every real number x, x²>x; q: For every real number x, either x>1 or x<1. b) "Mathematics is fun" - check whether this sentence is a statement.

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