Prepare the truth table for the following statement patterns
$\begin{align}
& \left( i \right)\left[ \left( p\to q \right)\wedge q \right]\to p \\
& \left( ii \right)\left( p\wedge q \right)\to \sim p \\
& \left( iii \right)\left( p\to q \right)\leftrightarrow \left( \sim p\vee q \right) \\
& \left( iv \right)\left( p\leftrightarrow r \right)\wedge \left( q\leftrightarrow p \right) \\
& \left( v \right)\left( p\vee \sim q \right)\to \left( r\wedge p \right) \\
\end{align}$
Answer
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Hint: To solve this question, we should know the basic truth tables. Truth table is a table which tells that the statement containing a combination of statements is true or false based on the elemental statements it is made of. The basic truth tables are p or q, p and q, p implies q, p double implies q and the notations are $p\vee q,p\wedge q,p\to q,p\leftrightarrow q$. There is another term called negation of a statement p and it is denoted as $\sim p$. In two elements p, q a total of four combinations are possible, which are both p and q are true, p is true and q is false, p is false and q is true, both p and q are false. The truth tables for the above statements are listed above. We shall denote true as T and false as F.
Using these tables, we can find the values of each of the statements asked in the question. For example let us consider $\left[ \left( p\to q \right)\wedge q \right]\to p$. We should first evaluate the value of $p\to q$ and then $p\to q$ as a single statement s and write the condition for $s\wedge q$ and then use it further. The main key is to take the process step by step evaluating the statements in the function which are in brackets.
Complete step by step answer:
In our question, we are asked different combinations of three statements p and q and r. First, we shall know the basic truth tables which are $p\vee q,p\wedge q,p\to q,p\leftrightarrow q$. Let us consider true as T and false as F.
Let us consider the statement $\left[ \left( p\to q \right)\wedge q \right]\to p$. To evaluate these kinds of complex statements, we should proceed step by step. First, we should evaluate $p\to q$ and then we should use this value to evaluate $\left( p\to q \right)\wedge q$, then the final statement.
So, finally, we can write that
Let us consider
$\left( p\wedge q \right)\to \sim p$
Finally, we can write that
Let us consider $\left( p\to q \right)\leftrightarrow \left( \sim p\vee q \right)$
Finally, we can write that
Let us consider $\left( p\leftrightarrow r \right)\wedge \left( q\leftrightarrow p \right)$
When there are three statements, a total of 8 combinations are possible. We shall evaluate the above statement for the possible 8 cases.
Finally, we can write that
Let us consider $\left( p\vee \sim q \right)\to \left( r\wedge p \right)$
We can write that
$\therefore $ Hence, we evaluated all the required statements.
Note: In these types of problems related to truth tables, the order in which we evaluate the statement plays a major role. The statements within the brackets should be evaluated first and then, we should proceed to the statements which are outside the brackets. For example, the truth value of the statement $\left( p\wedge q \right)\to \sim p$ is different from the statement $p\wedge \left( q\to \sim p \right)$ . The value changes when we change the order of evaluation. That is why the brackets have a higher importance in these statements.
| p | q | $\sim p$ | $p\vee q$ | $p\wedge q$ | $p\to q$ | $p\leftrightarrow q$ |
| T | T | F | T | T | T | T |
| T | F | F | T | F | F | F |
| F | T | T | T | F | T | F |
| F | F | T | F | F | T | T |
Using these tables, we can find the values of each of the statements asked in the question. For example let us consider $\left[ \left( p\to q \right)\wedge q \right]\to p$. We should first evaluate the value of $p\to q$ and then $p\to q$ as a single statement s and write the condition for $s\wedge q$ and then use it further. The main key is to take the process step by step evaluating the statements in the function which are in brackets.
Complete step by step answer:
In our question, we are asked different combinations of three statements p and q and r. First, we shall know the basic truth tables which are $p\vee q,p\wedge q,p\to q,p\leftrightarrow q$. Let us consider true as T and false as F.
| p | q | $\sim p$ | $p\vee q$ | $p\wedge q$ | $p\to q$ | $p\leftrightarrow q$ |
| T | T | F | T | T | T | T |
| T | F | F | T | F | F | F |
| F | T | T | T | F | T | F |
| F | F | T | F | F | T | T |
Let us consider the statement $\left[ \left( p\to q \right)\wedge q \right]\to p$. To evaluate these kinds of complex statements, we should proceed step by step. First, we should evaluate $p\to q$ and then we should use this value to evaluate $\left( p\to q \right)\wedge q$, then the final statement.
| p | q | $p\to q$ | $\left( p\to q \right)\wedge q$ | p | $\left[ \left( p\to q \right)\wedge q \right]\to p$ |
| T | T | T | T | T | T |
| T | F | F | F | T | T |
| F | T | T | T | F | F |
| F | F | T | F | F | T |
So, finally, we can write that
| p | q | $\left[ \left( p\to q \right)\wedge q \right]\to p$ |
| T | T | T |
| T | F | T |
| F | T | F |
| F | F | T |
Let us consider
$\left( p\wedge q \right)\to \sim p$
| p | q | $p\wedge q$ | $\sim p$ | $\left( p\wedge q \right)\to \sim p$ |
| T | T | T | F | F |
| T | F | F | F | T |
| F | T | F | T | T |
| F | F | F | T | T |
Finally, we can write that
| p | q | $\left( p\wedge q \right)\to \sim p$ |
| T | T | F |
| T | F | T |
| F | T | T |
| F | F | T |
Let us consider $\left( p\to q \right)\leftrightarrow \left( \sim p\vee q \right)$
| p | q | $\sim p$ | $p\to q$ | $\sim p\vee q$ | $\left( p\to q \right)\leftrightarrow \left( \sim p\vee q \right)$ |
| T | T | F | T | T | T |
| T | F | F | F | F | T |
| F | T | T | T | T | T |
| F | F | T | T | T | T |
Finally, we can write that
| p | q | $\left( p\to q \right)\leftrightarrow \left( \sim p\vee q \right)$ |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | T |
Let us consider $\left( p\leftrightarrow r \right)\wedge \left( q\leftrightarrow p \right)$
When there are three statements, a total of 8 combinations are possible. We shall evaluate the above statement for the possible 8 cases.
| p | q | r | $p\leftrightarrow r$ | $q\leftrightarrow p$ | $\left( p\leftrightarrow r \right)\wedge \left( q\leftrightarrow p \right)$ |
| T | T | T | T | T | T |
| T | T | F | F | T | F |
| T | F | T | T | F | F |
| T | F | F | F | F | F |
| F | T | T | F | F | F |
| F | T | F | T | F | F |
| F | F | T | F | T | F |
| F | F | F | T | T | T |
Finally, we can write that
| p | q | r | $\left( p\leftrightarrow r \right)\wedge \left( q\leftrightarrow p \right)$ |
| T | T | T | T |
| T | T | F | F |
| T | F | T | F |
| T | F | F | F |
| F | T | T | F |
| F | T | F | F |
| F | F | T | F |
| F | F | F | T |
Let us consider $\left( p\vee \sim q \right)\to \left( r\wedge p \right)$
| p | q | r | $\sim q$ | $p\vee \sim q$ | $r\wedge p$ | $\left( p\vee \sim q \right)\to \left( r\wedge p \right)$ |
| T | T | T | F | T | T | T |
| T | T | F | F | T | F | F |
| T | F | T | T | T | T | T |
| T | F | F | T | T | F | F |
| F | T | T | F | F | F | T |
| F | T | F | F | F | F | T |
| F | F | T | T | T | F | F |
| F | F | F | T | T | F | F |
We can write that
| p | q | r | $\left( p\vee \sim q \right)\to \left( r\wedge p \right)$ |
| T | T | T | T |
| T | T | F | F |
| T | F | T | T |
| T | F | F | F |
| F | T | T | T |
| F | T | F | T |
| F | F | T | F |
| F | F | F | F |
$\therefore $ Hence, we evaluated all the required statements.
Note: In these types of problems related to truth tables, the order in which we evaluate the statement plays a major role. The statements within the brackets should be evaluated first and then, we should proceed to the statements which are outside the brackets. For example, the truth value of the statement $\left( p\wedge q \right)\to \sim p$ is different from the statement $p\wedge \left( q\to \sim p \right)$ . The value changes when we change the order of evaluation. That is why the brackets have a higher importance in these statements.
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