Say the Opposite

Say the Opposite

Maths /

About Tier 2 - Relation

Requires a few ideas, quantities or steps to be connected.

Difficulty describes the challenge, not the solver. It is a Society guide rather than a formal academic level.

The tier is a Society guide to the amount of reasoning, technical work and insight a problem is expected to require. Your experience may vary depending on the topic and your background.

Logic & Proof / Quantifiers and Negation - 5-7 min

Consider a statement of this form: for every real $x$ , if $P(x)$ is true, then $Q(x)$ is true. Which option gives its logical negation for any choice of conditions $P$ and $Q$ ?

For example, take $P(x)$ to mean $x>2$ and $Q(x)$ to mean $x^2>4$ . The original statement then reads:

\text{For every }x\in\mathbb R,\quad x>2\;\Longrightarrow\;x^2>4.

Choose the rule that correctly negates the structure of the statement. The task is not to decide whether this particular example is true.

Assumptions

Use classical logic over the real numbers. P and Q are conditions on the same real x. Select a general negation rule, not a claim based only on the truth of the example.

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Collection question · Revision 1 · ID: logic-negation

Say the Opposite

Maths /

About Tier 2 - Relation

Requires a few ideas, quantities or steps to be connected.

Difficulty describes the challenge, not the solver. It is a Society guide rather than a formal academic level.

The tier is a Society guide to the amount of reasoning, technical work and insight a problem is expected to require. Your experience may vary depending on the topic and your background.

Say the Opposite

Logic & Proof / Quantifiers and Negation - 5-7 min

Consider a statement of this form: for every real $x$ , if $P(x)$ is true, then $Q(x)$ is true. Which option gives its logical negation for any choice of conditions $P$ and $Q$ ?

For example, take $P(x)$ to mean $x>2$ and $Q(x)$ to mean $x^2>4$ . The original statement then reads:

\text{For every }x\in\mathbb R,\quad x>2\;\Longrightarrow\;x^2>4.

Choose the rule that correctly negates the structure of the statement. The task is not to decide whether this particular example is true.

Assumptions

Use classical logic over the real numbers. P and Q are conditions on the same real x. Select a general negation rule, not a claim based only on the truth of the example.

Print Question Make an Editable Copy

Collection question · Revision 1 · ID: logic-negation