Answer
Verified390.3k+ views
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.
Recently Updated Pages
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE
Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE
What are the possible quantum number for the last outermost class 11 chemistry CBSE
Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE
Trending doubts
State the differences between manure and fertilize class 8 biology CBSE
Why are xylem and phloem called complex tissues aBoth class 11 biology CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
What would happen if plasma membrane ruptures or breaks class 11 biology CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
What precautions do you take while observing the nucleus class 11 biology CBSE
What would happen to the life of a cell if there was class 11 biology CBSE
Change the following sentences into negative and interrogative class 10 english CBSE