Let A={1,2,3,4,5} . Determine the truth value of each of the...

Let $A=\{1,2,3,4,5\} .$ Determine the truth value of each of the following statements: (a) $\quad(\exists x \in A)(x+3=10)$ (c) $(3 x \in A)(x+3<5)$ (b) $\quad(\forall x \in A)(x+3<10)$ (d) $\quad(\forall x \in A)(x+3 \leq 7)$ (a) False. For no number in $A$ is a solution to $x+3=10$. (b) True. For every number in $A$ satisfies $x+3<10$. (c) True. For if $x_{0}=1$, then $x_{0}+3<5$, i.e., 1 is a solution. (d) False. For if $x_{0}=5$, then $x_{0}+3$ is not less than or equal 7. In other words, 5 is not a solution to the given condition.

Let $A=\{1,2,3,4,5\} .$ Determine the truth value of each of the following statements:
(a) $\quad(\exists x \in A)(x+3=10)$
(c) $(3 x \in A)(x+3<5)$
(b) $\quad(\forall x \in A)(x+3<10)$
(d) $\quad(\forall x \in A)(x+3 \leq 7)$
(a) False. For no number in $A$ is a solution to $x+3=10$.
(b) True. For every number in $A$ satisfies $x+3<10$.
(c) True. For if $x_{0}=1$, then $x_{0}+3<5$, i.e., 1 is a solution.
(d) False. For if $x_{0}=5$, then $x_{0}+3$ is not less than or equal 7. In other words, 5 is not a solution to the given condition.

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Key Concepts

Predicate Logic

Predicate logic is a branch of mathematical logic that deals with predicates or properties that objects can have, and quantifiers which indicate the extent to which a predicate holds for a set of objects. It forms the basis for constructing meaningful statements about elements within a defined domain and assessing their truth based on given conditions.

Existential Quantification

Existential quantifiers state that there exists at least one element in the domain for which the predicate is true. In evaluating statements with an existential quantifier, the key approach is to identify whether at least one specific instance in the set satisfies the given condition.

Universal Quantification

Universal quantifiers assert that a predicate holds true for all elements in the specified domain. To validate such statements, each element in the domain must be examined; if even one element fails to satisfy the condition, the universal statement is false.

Counterexample Method

The counterexample method involves disproving a universal statement by finding a single element in the domain that does not meet the stated condition. In logical assessments, providing a counterexample effectively demonstrates that the universal condition does not hold for the entire set.

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