Answer
Verified399k+ views
Hint: This is a problem of mathematical reasoning. We will start by writing each of these statements in their symbolic forms. We will then simplify these symbolic forms. Finally, we will write the truth table for each of these statements and check which statements form the same truth table.
Complete step-by-step answer:
We have been given two statements- p: it is raining and q: weather is humid. We will first try to write these in a symbolic form.
If it is not raining then the weather is not humid = $\text{~}p \to \text{~}q$
It is raining if and only if the weather is humid = $p \leftrightarrow q$
It is not true that it is not raining or the weather is humid = $\text{~}\left( {\text{~}p \vee q} \right)$
It is raining but the weather is not humid = $p \wedge \text{~}q$
We have the symbolic representation of these statements, so we will now try to form the truth table of all these statements. The statements which give the same truth table are logically equivalent, and this will give us our answer.
A. If it is not raining then the weather is not humid.
B. It is raining if and only if the weather is humid.
C. It is not true that it is not raining or the weather is humid.
D. It is raining but the weather is not humid.
From these four truth tables, it is clearly visible that the tables formed by options C and D are logically equivalent, and hence the correct options are C and D.
Note: The important thing for students to do here is that they need to memorize the meanings of all the symbols, because they will help in forming the truth tables for each of the statements. For example, in this problem, ~ stands for ‘not’, ∧ stands for ‘and’, ∨ stands for ‘or’ and so on.
Complete step-by-step answer:
We have been given two statements- p: it is raining and q: weather is humid. We will first try to write these in a symbolic form.
If it is not raining then the weather is not humid = $\text{~}p \to \text{~}q$
It is raining if and only if the weather is humid = $p \leftrightarrow q$
It is not true that it is not raining or the weather is humid = $\text{~}\left( {\text{~}p \vee q} \right)$
It is raining but the weather is not humid = $p \wedge \text{~}q$
We have the symbolic representation of these statements, so we will now try to form the truth table of all these statements. The statements which give the same truth table are logically equivalent, and this will give us our answer.
A. If it is not raining then the weather is not humid.
| p | q | ~p | ~q | ∼p ∨ ∼q | ~p→∼q |
| T | T | F | F | T | T |
| T | F | F | T | F | T |
| F | T | T | F | T | F |
| F | F | T | T | T | T |
B. It is raining if and only if the weather is humid.
| p | q | ~p | ~q | ∼p ∨ ∼q | p↔q |
| T | T | F | F | T | T |
| T | F | F | T | F | F |
| F | T | T | F | T | F |
| F | F | T | T | T | T |
C. It is not true that it is not raining or the weather is humid.
| p | q | ~p | ~q | ∼p ∨ ∼q | ∼(~p ∨ q) |
| T | T | F | F | T | F |
| T | F | F | T | F | T |
| F | T | T | F | T | F |
| F | F | T | T | T | F |
D. It is raining but the weather is not humid.
| p | q | ~p | ~q | ∼p ∨ ∼q | p ∧ ∼q |
| T | T | F | F | T | F |
| T | F | F | T | F | T |
| F | T | T | F | T | F |
| F | F | T | T | T | F |
From these four truth tables, it is clearly visible that the tables formed by options C and D are logically equivalent, and hence the correct options are C and D.
Note: The important thing for students to do here is that they need to memorize the meanings of all the symbols, because they will help in forming the truth tables for each of the statements. For example, in this problem, ~ stands for ‘not’, ∧ stands for ‘and’, ∨ stands for ‘or’ and so on.
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
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
What is BLO What is the full form of BLO class 8 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The cell wall of prokaryotes are made up of a Cellulose class 9 biology CBSE
What organs are located on the left side of your body class 11 biology CBSE
Select the word that is correctly spelled a Twelveth class 10 english CBSE
a Tabulate the differences in the characteristics of class 12 chemistry CBSE