discrete_mathematics_with_applications/chapter_3/notes.md

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Definition

A predicate is a sentence that contains a finite number of variables and becomes a statement when specific values are substituted for the variables. The domain of a predicate variable is the set of all values that may be substituted in place of the variable.

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Definition

If P(x) is a predicate and x has domain D, the truth set of P(x) is the set of all elements of D that make P(x) true when they are substituted for x. The truth set of P(x) is denoted

 \{x \in D | P(x)\} 

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Definition

Let Q(x) be a predicate and D the domain of x. A universal statement is a statement of the form "\forall x \in D, Q(x)." It is defined to be true if, and only if, Q(x) is true for each individual x in D. It is defined to be false if, and only if, Q(x) is false for at least one x in D. A value for x for which Q(x) is false is called a counterexample to the universal statement.

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Definition

Let Q(x) be a predicate and D the domain of x. An existential statement is a statement of the form "\exists x \in D such that Q(x)." It is defined to be true if, and only if, Q(x) is true for at least one x in D. It is false if, and only if, Q(x) is false for all x in D.

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Notation

Let P(x) and Q(x) be predicates and suppose the domain of x is D.

• The notation P(x) \Rightarrow Q(x) means that every element in the truth set of P(x) is in the truth set of Q(x), or, equivalently, \forall x, P(x) \to Q(x).

• The notation P(x) \Leftrightarrow Q(x) means that P(x) and Q(x) have identical truth sets, or, equivalently, \forall x, P(x) \leftrightarrow Q(x).

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Theorem 3.2.1 Negation of a Universal Statement

The negation of a statement of the form

 \forall \text{ in } D, Q(x) 

is logically equivalent to a statement of the form

 \exists \text{ in } D \text{ such that } \neg Q(x) 

Symbolically,

 \neg(\forall x \in D, Q(x)) \equiv \exists x \in D \text{ such that } \neg Q(x) 

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Theorem 3.2.2 Negation of an Existential Statement

The negation of a statement of the form

 \exists \text{ in } D \text{ such that } Q(x) 

is logically equivalent to a statement of the form

 \forall x \text{ in } D, \neg Q(x) 

Symbolically,

 \neg(\exists x \in D \text{ such that } Q(x)) \equiv \forall x \in D, \neg Q(x) 

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Negation of a Universal Conditional Statement

 \neg(\forall x, \text{ if } P(x) \text{ then } Q(x)) \equiv \exists x \text{ such that } P(x) \text{ and } \neg Q(x) 

 \neg(\forall x, P(x) \to Q(x)) \equiv \exists x, (P(x) \wedge \neg Q(x)) 

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Definition

Consider a statement of the form \forall x \in D, \text{ if } P(x) \text{ then } Q(x).

1. Its contrapositive is the statement \forall x \in D, \text{ if } \neg Q(x) \text{ then } \neg P(x).

2. Its converse is the statement \forall x \in D, \text{ if } Q(x) \text{ then } P(x).

3. Its inverse is the statement \forall x \in D, \text{ if } \neg P(x) \text{ then } \neg Q(x).

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Definition

• "\forall x, r(x) is a sufficient condition for $s(x)$" means "\forall x, \text{ if } r(x) \text{ then } s(x)."

• "\forall x, r(x) is a necessary condition for $s(x)$" means "$\forall x, \text{ if } \neg r(x) \text{ then } \neg s(x)$" or, equivalently, "\forall x, \text{ if } s(x) \text{ then } r(x)."

• "\forall x, r(x) only if $s(x)$" means "$\forall x, \text{ if } \neg s(x) \text{ then } \neg r(x)$" or, equivalently, "\forall x, \text{ if } r(x) \text{ then } s(x)."