Question

The statement that is true among the following is-
A.$p \Rightarrow q$ is equivalent to $p \wedge - q$
B.$p \vee q$ and $p \wedge q$ have the same truth value
C.The converse of$\tan x = 0 \Rightarrow x = 0$ is $x \ne 0 \Rightarrow \tan x = 0$
D.The contra-positive of $3x + 2 = 8 \Rightarrow x = 2$ is $x \ne 2 \Rightarrow 3x + 2 \ne 8$

Answer 1

We can check each statement one by one to know which statement is true. For the first statement we will draw the truth table. If both of the relations truth tables are the same then the statement is true. For the second statement we will use the definition of conjunction and disjunction. Both are opposite to each other so their truth table does not have the same truth values. We know that the converse of $p \Rightarrow q$ is $q \Rightarrow p$ so for the third statement, we will check if this applies to that statement or not. To check fourth statement we will check whether fourth statement gives- For $p \Rightarrow q$, the contra-positive is $- p \Rightarrow - q$

Complete step-by-step answer:
We have to find which statement is true so we will check each statement-
We will first check option A-
The statement is “$p \Rightarrow q$ is equivalent to $p \wedge - q$”
For this we will draw the truth table-
For p, we will choose consecutively two truths and two false. For q, we will choose alternate true and false. The truth table is-

p	q	$- q$	$p \Rightarrow q$	$p \wedge - q$
T	T	F	T	F
T	F	T	F	T
F	T	F	T	F
F	F	T	T	F

Here we can see that the last two tables have just opposite values so it is a contradiction and not a tautology. So option A is incorrect.
We will check option B-
The statement is “$p \vee q$ and $p \wedge q$ have the same truth value”.
We know that$\vee$ is the sign of disjunction in which a proposition is true when either one or both of p and q are true and is false when both p and q are false.
But$\wedge$ is the sign of conjunction in which a proposition is true when both p and q are true and is false when either or both of p and q are false.
Hence option B is incorrect.
We will check option C-
The statement is “The converse of $\tan x = 0 \Rightarrow x = 0$ is $x \ne 0 \Rightarrow \tan x = 0$”
Let the statement $\tan x = 0$ is p and $x = 0$ is q. So here it is given-
For $p \Rightarrow q$ the converse is$- p \Rightarrow - q$ . This is a contra-positive statement
And we know that the converse of $p \Rightarrow q$ is $q \Rightarrow p$ . Hence option C is also incorrect.
Now we will check option D-
The statement is-The contra-positive of $3x + 2 = 8 \Rightarrow x = 2$ is $x \ne 2 \Rightarrow 3x + 2 \ne 8$
Let $3x + 2 = 8$is p and $x = 2$is q. Then we can write the given statement as-
For $p \Rightarrow q$, the contra-positive is $- p \Rightarrow - q$
So this statement is true.
Hence the correct answer is D.
Note: We can also check the option A by this method-
We know that $p \Rightarrow q$ is equivalent to $\left( { \sim p \vee q} \right)$ and We also know that \sim \left( {p \Rightarrow q} \right) \equiv \left\\{ {p \wedge \sim q} \right\\}
Hence the truth tables of $p \Rightarrow q$and $p \wedge - q$ will not be equivalent. So option A is incorrect.