# Truth Table for P(x) & R(x), ~Q(x) & P(x) in U

• MHB
• trevor
In summary, the conversation discussed open propositions P(x), Q(x), and R(x) over the universe U = {− 4,−2, 0, 1, 3, 5, 6, 8, 10}. It also considered finding the truth-values for the compound propositions P(x) ∧ R(x) and [~ Q(x)] ∧ P(x) on a single truth table. The truth table, using 0 for false and 1 for true, shows that P(x) ∧ R(x) is only true for x = 5 while [~ Q(x)] ∧ P(x) is true for all values of x except x = 5.
trevor
Consider the following open propositions over the universe U = {− 4,−2, 0, 1, 3, 5, 6,
8, 10}
P(x): x ≥ 4
Q(x): x 2 = 25
R(x): s is a multiple of 2
Find on a single truth table the truth-values of the following.
i. P(x ) ∧ R (x )
ii. [~ Q(x )] ∧ P(x )]

Please note that according to https://mathhelpboards.com/rules/ 11 you are expected to explain your attempts at solving the problem or describe your difficulty. At the very least please make sure that the problem statement is typed correctly and not simply copied, which results in formulas like "x 2 = 25", which don't make sense. And what is $s$ in "$s$ is a multiple of 2" since it is supposed to be the definition of $R(x)$?

Truth tables are not usually used in predicate logic, but since the universe is finite, their use makes sense here. I assume the table should look like this (I use 0 for false and 1 for true).
$\begin{array}{r|c|c|c|c|c} x & P(x) & Q(x) & R(x) & P(x)\land R(x) & \neg Q(x)\land P(x)\\ \hline -4&0&&1&0&\\ -2&0&&1&0&\\ 0&0&&1&0&\\ 1&0&&0&0&\\ 3&0&&0&0&\\ 5&1&&&&\\ 6&&&&&\\ 8&&&&&\\ 10&&&&& \end{array}$
Try filling the rest of the table.

## What is a truth table?

A truth table is a table that displays all the possible combinations of values for a propositional statement or set of statements, and shows the resulting truth value for each combination.

## What is P(x) & R(x) in the given truth table?

P(x) & R(x) is a compound statement that represents the logical AND operation between two statements, P(x) and R(x). It will only be true if both P(x) and R(x) are true.

## What does ~Q(x) represent in the truth table?

~Q(x) is a negated statement that represents the logical NOT operation on the statement Q(x). It will be true if Q(x) is false, and false if Q(x) is true.

## What is the significance of "U" in the truth table?

"U" in the truth table refers to the universe of discourse, which is the set of all possible values that a variable can take. In this case, it represents the domain for the variables P(x), R(x), and Q(x).

## What is the purpose of using a truth table?

The purpose of a truth table is to systematically and clearly organize all the possible combinations of values for a logical statement or set of statements, and to determine the resulting truth value for each combination. This helps to analyze and understand the logical relationships between different statements and to evaluate their validity.

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