Predicates

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Predicate: a logical statement or function that contains variables and becomes either true or false when the variables are assigned specific values.

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Example: Consider the statement: "x is greater than 3." This can be represented as a predicate:P(x): x > 3 The truth value of P(x) depends on the value of x. If x = 5, P(x) is true, but if x = 2, P(x) is false.

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Domain of Discourse: the set of all possible values that the variables in a predicate can take. For instance, if the domain is the set of all integers, P(x) is evaluated for every integer.

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Quantifiers

Quantifiers allow us to make statements about entire sets of elements rather than individual elements. There are two main types of quantifiers: universal and existential.

Universal Quantifier ( ∀ )

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The universal quantifier ∀: "for all" or "for every." It indicates that a predicate is true for every element in its domain.

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Example:∀x P(x) means "For all x, P(x) is true." If P(x): x > 3 and the domain is the set of positive integers, then ∀x P(x) is false because not all positive integers are greater than 3.
Notation:

The universal quantifier is written as ∀x, followed by the predicate. It can also be expressed as a conditional statement:

∀x (P(x) → Q(x)) means "For every x, if P(x) is true, then Q(x) is true."

Existential Quantifier ( ∃ )

Definition:

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The existential quantifier ∃: means "there exists" or "for some." It indicates that there is at least one element in the domain for which the predicate is true.

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- Example:∃x P(x) means "There exists an x such that P(x) is true." If P(x): x > 3 and the domain is the set of integers, then ∃x P(x) is true because there are integers greater than 3 (e.g., 4, 5, etc.).
Notation:

The existential quantifier is written as ∃x, followed by the predicate:

∃x P(x) means that at least one value of x in the domain makes P(x) true.

Combining Quantifiers

In many mathematical statements, we use both universal and existential quantifiers to express complex ideas.

Example: "For every positive integer x, there exists a y such that y = x + 1." This can be written as:∀x ∃y (y = x + 1) This means that for each x, you can find a y that satisfies the equation.

Quantifiers can also be nested, and the order of the quantifiers matters. For instance,