Determine the truth value of each of these statements if the...

Determine the truth value of each of these statements if the domain of each variable consists of all real numbers. $$ \begin{array}{ll}{\text { a) } \forall x \exists y\left(x^{2}=y\right)} & {\text { b) } \forall x \exists y\left(x=y^{2}\right)} \\ {\text { c) } \exists x \forall y(x y=0)} & {\text { d) } \exists x \exists y(x+y \neq y+x)}\end{array} $$ $$ \begin{array}{l}{\text { e) } \forall x(x \neq 0 \rightarrow \exists y(x y=1))} \\ {\text { f) } \exists x \forall y(y \neq 0 \rightarrow x y=1)} \\ {\text { g) } \forall x \exists y(x+y=1)} \\ {\text { h) } \exists x \exists y(x+2 y=2 \wedge 2 x+4 y=5)} \\ {\text { i) } \forall x \exists y(x+y=2 \wedge 2 x-y=1)} \\ {\text { j) } \forall x \forall y \exists z(z=(x+y) / 2)}\end{array} $$

Determine the truth value of each of these statements if the domain of each variable consists of all real numbers.
$$
\begin{array}{ll}{\text { a) } \forall x \exists y\left(x^{2}=y\right)} & {\text { b) } \forall x \exists y\left(x=y^{2}\right)} \\ {\text { c) } \exists x \forall y(x y=0)} & {\text { d) } \exists x \exists y(x+y \neq y+x)}\end{array}
$$
$$
\begin{array}{l}{\text { e) } \forall x(x \neq 0 \rightarrow \exists y(x y=1))} \\ {\text { f) } \exists x \forall y(y \neq 0 \rightarrow x y=1)} \\ {\text { g) } \forall x \exists y(x+y=1)} \\ {\text { h) } \exists x \exists y(x+2 y=2 \wedge 2 x+4 y=5)} \\ {\text { i) } \forall x \exists y(x+y=2 \wedge 2 x-y=1)} \\ {\text { j) } \forall x \forall y \exists z(z=(x+y) / 2)}\end{array}
$$

Key Concepts

Systems of Equations

Systems of equations involve multiple equations that must be satisfied simultaneously. Determining the truth value of statements based on systems requires assessing the consistency and solvability of the equations when taken together. It often involves techniques to either find a solution that satisfies all conditions or to prove that no such solution exists.

Universal Quantifier

The universal quantifier (?) indicates that a statement must hold for every element in the domain. When working with predicates over the set of real numbers, understanding how to validate or refute a universally quantified statement is essential. One must ensure that the stated property is true for every possible choice of the variable within the given domain.

Existential Quantifier

The existential quantifier (?) asserts that there is at least one element in the domain for which the predicate holds. In the context of real numbers, demonstrating an existential statement involves finding a specific example, or proving by construction, where the condition is met.

Logical Implication

Logical implication (?) is used to connect propositions in the format 'if... then...'. In mathematical logic, it plays a crucial role, especially in conditional statements. Understanding how to work with implications is important when evaluating statements involving conditions, such as the existence of an inverse for non-zero numbers.

Domain of Real Numbers

Restricting the domain to the set of real numbers impacts the truth values of statements by limiting the available elements that can be used to satisfy the predicates. Many properties such as the existence of square roots, multiplicative inverses, and the structure of algebraic operations rely on the characteristics of the real numbers.

Properties of Squares and Square Roots

Operations involving squaring (raising to the power of two) and taking square roots are critical because they have inherent restrictions in the real numbers; for instance, every real number squared is non-negative, and not every real number is the square of another real number. This concept is particularly important when evaluating predicates that involve equations like x² = y or x = y².

Multiplicative Inverse

The concept of a multiplicative inverse in the real numbers states that for every non-zero real number x, there exists a real number y such that x · y = 1. This property is foundational when analyzing statements that require the existence of reciprocals, and it is closely linked with the conditions set by non-zero constraints.

Commutative Property

The commutative property of addition and multiplication states that the order in which two numbers are added or multiplied does not affect the result (i.e., x + y = y + x and x · y = y · x). Understanding this property helps in evaluating statements that test whether operations like addition are commutative or assert the existence of exceptions.

Transcript

00:03 So we have for all x to exist and y, such that x squared is equal to y.

00:10 Well, is the statement always true? yeah, it's true because, let's see.

00:18 We can just find, so let's say y is 2, then we can have an x that is, or if x is 1, then our y is 2, then our y is 4.

00:33 Is solving an equation based on where x is our y is that value squared for parts b we have that for all x there should exist some y such that x is equal to y squared okay well let's see if you think about this what about square roots of negative values well it isn't possible for a negative real number because the real square roots does not exist just the statement is false.

01:22 To part c, we have exist some x for all at y, such that x times y is equal to 0.

01:29 Ok, so this statement is true, because then what if, well, x can always equal 0.

01:35 0 times any number is always 0.

01:37 So part d, we have there is x and y, which that x plus y is not equal to y plus x.

01:52 Well, we know that the property of addition is commutative, so these two should be able to hold.

02:09 So this statement is all this meaningful.

02:13 Now we have e, which is for all x, there exists on x not equal to zero, such that this implies that there exists in y, such that x times y is equal to 1.

02:29 Well, yeah, so basically whatever value we have for x, we just take the inverse of x and let that equal to y, and then x times our inverse of x is equal to 1.

02:41 So this is true.

02:43 For f, we have there exists some x for all y, such that if y is not equal to fell, this implies that x comma y is equal to 1.

02:57 Okay, so this statement says that there exists a real number such that for every non -zero real number, their product equals 1, but this is obviously not true.

03:10 So the statement is felt...

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