Mathematical Statement

Abstract

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• Also known as Proposition
• 3 important types are Universal, Conditional & Existential
• We can form statements that are Universal & Conditional, or more to form complex statements
• Either True or False

Warning

• Doesn’t depend on any Variable, unless it is Logical Equivalence equation
• Not in a question form

Different Types of Statements

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Universal

• Made of Symbol, Variable & Quantifiers

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

• 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

Existential

• Made of Symbol, Variable & Quantifiers

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

• true iff Q(x) is true for at least one x in D
• false iff Q(x) is false for all x in D
• For simple ones, usually existential comes with Conjunction

Negation of Universal

• Logical Equivalence to Existential
• There is AT LEAST ONE that IS NOT

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

Negation of Existential

• Logical Equivalence to Universal
• ALL ARE NOT

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

Negation of Universal Conditional

• Make use of Negation of Universal & Implication Law

Universal Conditional

• Made of Symbol, Variable & Quantifiers & Conditional Statement

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

• Can be simplified to Universal

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

Conditional

• Conditional Statement
• If…, then…

Terminologies

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Statement

• A sentence that is True or False, but not both
• Can be presented with variables like p, q, r & s etc
• The tip here is to keep it atomic, especially in Mathematical Proof that is complicated

Logical Connectives

• Negation: NOT performed first
• Disjunction: OR ,Conjunction: AND coequal, make sure proper parentheses is used to avoid Ambiguous
• If-then/implies: performed last, coequal with iff

Statement/Propositional Form

• Make up with Statement variables & Logical Connectives

Compound Statement

• Make up with Statement variables & Logical Connectives

Conditional Statement, convert it using Implication Law to make it much less confusing

Tautology t

• Statement/Propositional Form that is always true

Contradiction c

• Statement/Propositional Form that is always false

Implicitly Quantified

• The Quantifiers are assumed without specified explicitly

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

Multiply-Quantified

• Statement with more than one Quantifiers