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).