Consider the open sentence Q(a, b): a-b<0. where the domain...

Consider the open sentence $Q(a, b): a-b<0$. where the domain of $a$ is $A=\{3,5,8\}$ and the domain of $b$ is $B=\{3,6,10\}$.
(a) State the quantified statement $\exists b \in B, \forall a \in A, Q(a, b)$ in words.
(b) Show the quantified statement in (a) is true.

Key Concepts

Quantifiers

Quantifiers are symbols used in logic to indicate the extent to which a predicate is asserted over a domain. The existential quantifier (?) states that there is at least one element in the domain for which the predicate is true, while the universal quantifier (?) asserts that the predicate holds for every element in the domain.

Domain of Discourse

The domain of discourse is the set of values that the variables in a logical statement can take. It provides the context for evaluating the truth of quantified statements by defining the eligible elements for the variables involved.

Open Sentences or Predicates

An open sentence, or predicate, is a statement that contains variables and becomes a complete proposition only when specific values are substituted for those variables. It expresses a property or relation that can be true or false depending on the variable assignments.

Order Relations

Order relations, such as inequalities, are mathematical expressions that compare matters such as size or order between elements. They are an essential part of logical reasoning, providing criteria for evaluating whether one element is less than, greater than, or equal to another.

Statement Verification

Statement verification in the context of quantified logic involves demonstrating that the logical structure holds by substituting specific elements from the domains into the predicate. This process checks that the conditions imposed by the quantifiers are satisfied, often by finding a specific instance or by showing a condition holds across the entire domain.

Please give Ace some feedback