Universal Statement

Universal Statement

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$$\forall x\in D,Q(x)$$

• Made of Predicate Symbol, Predicate Variable & Quantifiers
• true iff Q(x) is true for every x in D
• false iff Q(x) is false for at least one x in D
• For simple ones, usually universal comes with Conditional Statement

Negation of Universal

$$\neg (\forall x\in D,Q(x))\equiv \exists x\in D,\neg Q(x)$$

• Negation of Universal is Logical Equivalence to Existential Statement
• The above Predicate means There is AT LEAST ONE that IS NOT

Universal Conditional

$$\forall x(ifP(x)\to Q(x))$$

• Made of Universal Statement & Conditional Statement

  Simplification \forall x \in P, Q(x)

  The above statement can be reduced to

Negation of Universal Conditional

• Make use of Negation of Universal & Implication Law

Vacuous Truth of Universal

• Given the statement: All balls in the bowl are blue, however no balls in the bowl. The statement is vacuously true, because the Negation of Universal is One of the balls in the bowl isn't blue which is obviously false

Vacuous Truth of Universal Conditional

• Given $\forall x\in D,P(x)\to Q(x)$
• It is Vacuously True if and only if P(x) is false for every x in D
• Vacuous Truth of Universal also applies here for the Quantifiers part