In each of the following, two open sentences P(x, y) and...

In each of the following, two open sentences $P(x, y)$ and $Q(x, y)$ are given, where the domain of both $x$ and $y$ is $\mathbf{Z}$. Determine the truth value of $P(x, y) \Leftrightarrow Q(x, y)$ for the given values of $x$ and $y$. (a) $P(x, y): x^2-y^2=0$. and $Q(x, y): x=y$. $(x, y) \in\{(1,-1),(3,4),(5,5)\}$. (b) $P(x, y):|x|=|y|$. and $Q(x, y): x=y$. $(x, y) \in\{(1,2),(2,-2),(6,6)\}$. (c) $P(x, y): x^2+y^2=1$. and $Q(x, y): x+y=1$. $(x, y) \in\{(1,-1),(-3,4),(0,-1),(1,0)\}$.

In each of the following, two open sentences $P(x, y)$ and $Q(x, y)$ are given, where the domain of both $x$ and $y$ is $\mathbf{Z}$. Determine the truth value of $P(x, y) \Leftrightarrow Q(x, y)$ for the given values of $x$ and $y$.
(a) $P(x, y): x^2-y^2=0$. and $Q(x, y): x=y$. $(x, y) \in\{(1,-1),(3,4),(5,5)\}$.
(b) $P(x, y):|x|=|y|$. and $Q(x, y): x=y$. $(x, y) \in\{(1,2),(2,-2),(6,6)\}$.
(c) $P(x, y): x^2+y^2=1$. and $Q(x, y): x+y=1$. $(x, y) \in\{(1,-1),(-3,4),(0,-1),(1,0)\}$.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Open Sentences

Open sentences, also known as predicates, are mathematical expressions that contain variables; they become complete statements with truth values only after the variables are assigned specific values. They are fundamental in studying logical relationships and in forming predicate formulas.

Logical Implication

Logical implication, denoted by the symbol '?', is a logical connective that asserts if the antecedent (the left-hand statement) is true, then the consequent (the right-hand statement) must also be true for the whole expression to be true. If the antecedent is false, the implication is considered true by the nature of material implication.

Truth Value

The truth value of a statement indicates whether it is true or false. In predicate logic, the truth value of an open sentence is determined by the specific assignment of values to the variables from its domain, and in implications, the truth value depends on the interaction between the antecedent and the consequent.

Domain of Discourse

The domain of discourse is the set of all possible elements that the variables in an expression or statement can take. It frames the context in which predicates and logical implications are evaluated, ensuring that variable assignments are valid and meaningful within that set.

Transcript

Please give Ace some feedback