Samacheer Kalvi Class 12 Maths Solutions Chapter 12 Discrete Mathematics Exercise 12.2

 

Question 1. Let p : Jupiter is a planet and q: India is an island be any two simple statements. Give verbal sentence describing each of the following statements.
(i) \( \neg P \)
(ii) \( P \wedge \neg q \)
(iii) \( \neg p \vee q \)
(iv) \( p \rightarrow \neg q \)
(v) \( p \leftrightarrow q \)
Answer:
(i) \( \neg P \): Jupiter is not a planet.
(ii) \( P \wedge \neg q \): Jupiter is a planet and India is not an island.
(iii) \( \neg p \vee q \): Jupiter is not a planet or India is an island.
(iv) \( p \rightarrow \neg q \): If Jupiter is a planet, then India is not an island.
(v) \( p \leftrightarrow q \): Jupiter is a planet if and only if India is an island. These statements show how we can use logic symbols to represent everyday sentences in a more concise way.
In simple words: We take the given statements about Jupiter and India and translate them into full sentences based on the logic symbols like "not", "and", "or", "if-then", and "if and only if".

🎯 Exam Tip: Remember what each logical connective (like \( \neg \), \( \wedge \), \( \vee \), \( \rightarrow \), \( \leftrightarrow \)) means in plain English to correctly translate between symbolic and verbal forms.

 

Question 2. Write each of the following sentences in symbolic form using statement variables p and q.
(i) 19 is not a prime number and all the angles of a triangle are equal.
(ii) 19 is a prime number or all the angles of a triangle are not equal.
(iii) 19 is a prime number and all the angles of a triangle are equal.
(iv) 19 is not a prime number.
Answer:
Let p: 19 is a prime number
Let q: All the angles of a triangle are equal
(i) 19 is not a prime number and all the angles of a triangle are equal \( \implies \neg p \wedge q \)
(ii) 19 is a prime number or all the angles of a triangle are not equal \( \implies p \vee \neg q \)
(iii) 19 is a prime number and all the angles of a triangle are equal \( \implies p \wedge q \)
(iv) 19 is not a prime number \( \implies \neg p \). Symbolic logic helps us simplify complex sentences into clear mathematical expressions for easier analysis.
In simple words: We change sentences into math symbols. "Not" becomes \( \neg \), "and" becomes \( \wedge \), "or" becomes \( \vee \). We use 'p' for "19 is a prime number" and 'q' for "all angles of a triangle are equal" to build the symbolic form.

🎯 Exam Tip: Clearly define your simple statements (p, q) first before converting the compound sentences into symbolic form. Pay close attention to keywords like "not", "and", "or" to choose the correct logical connectives.

 

Question 3. Determine the truth value of each of the following statements
(i) If 6 + 2 = 5, then the milk is white.
(ii) China is in Europe or \( \sqrt{3} \) is an integer.
(iii) It is not true that 5 + 5 = 9 or Earth is a planet.
(iv) 11 is a prime number and all the sides of a rectangle are equal.
Answer:
(i) If 6 + 2 = 5, then the milk is white.
Let p: 6 + 2 = 5 (False)
Let q: Milk is white (True)
The statement is \( p \rightarrow q \). Since p is False and q is True, \( F \rightarrow T \) is True.
\( \implies \) So, \( p \rightarrow q \) has the truth value True.

(ii) China is in Europe or \( \sqrt{3} \) is an integer.
Let p: China is in Europe (False)
Let q: \( \sqrt{3} \) is an integer (False)
The statement is \( p \vee q \). Since p is False and q is False, \( F \vee F \) is False.
\( \implies \) So, \( p \vee q \) has the truth value False.

(iii) It is not true that 5 + 5 = 9 or Earth is a planet.
Let p: 5 + 5 = 9 (False)
Let q: Earth is a planet (True)
The statement is \( \neg p \vee q \). Since \( \neg p \) is True and q is True, \( T \vee T \) is True.
\( \implies \) So, \( \neg p \vee q \) has the truth value True.

(iv) 11 is a prime number and all the sides of a rectangle are equal.
Let p: 11 is a prime number (True)
Let q: All the sides of a rectangle are equal (False)
The statement is \( p \wedge q \). Since p is True and q is False, \( T \wedge F \) is False.
\( \implies \) So, \( p \wedge q \) has the truth value False. When evaluating truth values, remember that an implication is only false if a true statement implies a false one.
In simple words: We check if each small part of the statement is true or false. Then we use rules for "if-then", "or", "not", and "and" to find the truth value of the whole statement. For example, "if false then true" is always true.

🎯 Exam Tip: Break down each compound statement into simple propositions and determine their individual truth values. Then apply the rules of logical connectives step-by-step to find the overall truth value.

 

Question 4. Which one of the following sentences is a proposition?
1. 4 + 7 = 12
2. What are you doing?
3. \( 3^n \le 81 \), \( n \in N \)
4. Peacock is our national bird.
5. How tall this mountain is!
Answer:
1. \( 4 + 7 = 12 \) is a proposition (it is false).
2. "What are you doing?" is not a proposition (it is a question).
3. \( 3^n \le 81 \), \( n \in N \) is a proposition (it can be true or false depending on 'n').
4. "Peacock is our national bird." is a proposition (it is true).
5. "How tall this mountain is!" is not a proposition (it is an exclamation). A proposition is a statement that is either definitively true or definitively false, and cannot be both at the same time.
In simple words: A proposition is a sentence that can be clearly said to be true or false, not a question or an exclamation. From the list, numbers 1, 3, and 4 are propositions because we can say if they are true or false.

🎯 Exam Tip: A proposition must be a declarative sentence that can be assigned a truth value (True or False). Questions, commands, and exclamations are not propositions.

 

Question 5. Write the converse, inverse, and contrapositive of each of the following implication.
(i) If x and y are numbers such that \( x = y \), then \( x^2 = y^2 \)
(ii) If a quadrilateral is a square then it is a rectangle.
Answer:
(i) If x and y are numbers such that \( x = y \), then \( x^2 = y^2 \).
Converse: If x and y are numbers such that \( x^2 = y^2 \), then \( x = y \).
Inverse: If x and y are numbers such that \( x \neq y \), then \( x^2 \neq y^2 \).
Contrapositive: If x and y are numbers such that \( x^2 \neq y^2 \), then \( x \neq y \).

(ii) If a quadrilateral is a square then it is a rectangle.
Converse: If a quadrilateral is a rectangle, then it is a square.
Inverse: If a quadrilateral is not a square, then it is not a rectangle.
Contrapositive: If a quadrilateral is not a rectangle, then it is not a square. Understanding these transformations helps us analyze the logical connections between statements.
In simple words: For a statement "If A, then B":
- The converse is "If B, then A."
- The inverse is "If not A, then not B."
- The contrapositive is "If not B, then not A."

🎯 Exam Tip: Remember the definitions: Converse swaps the hypothesis and conclusion. Inverse negates both. Contrapositive swaps and negates both. The contrapositive always has the same truth value as the original statement.

 

Question 6. Construct the truth table for the following statements.
(i) \( \neg P \wedge \neg q \)
(ii) \( \neg (P \wedge \neg q) \)
(iii) \( (p \vee q) \vee \neg q \)
(iv) \( (\neg p \rightarrow r) \wedge (p \leftrightarrow q) \)
Answer:
(i) \( \neg P \wedge \neg q \)

(ii) \( \neg (P \wedge \neg q) \)

(iii) \( (p \vee q) \vee \neg q \)

(iv) \( (\neg p \rightarrow r) \wedge (p \leftrightarrow q) \)
Truth tables help us see all possible truth values for complex logical statements given the truth values of their basic parts. It is a systematic way to analyze logic.
In simple words: We make tables to show if a statement is true (T) or false (F) for every possible combination of true or false inputs for P, Q, and R. We fill in the columns step-by-step using the rules for "not", "and", "or", "if-then", and "if and only if".

🎯 Exam Tip: When constructing truth tables for multiple variables, ensure you list all \( 2^n \) possible combinations of truth values for 'n' simple statements. Work systematically, building columns for sub-expressions before the final compound statement.

 

Question 7. Verify whether the following compound propositions are tautologies or contradictions or contingency.
(i) \( (P \wedge q) \wedge \neg (p \vee q) \)
(ii) \( ((P \vee q) \wedge \neg p) \rightarrow q \)
(iii) \( (p \rightarrow q) \rightarrow (\neg p \rightarrow q) \)
(iv) \( ((p \rightarrow q) \wedge (q \rightarrow r)) \rightarrow (p \rightarrow r) \)
Answer:
(i) \( (P \wedge q) \wedge \neg (p \vee q) \)

The entries in the last column are all False.
\( \implies \) The given statement is a contradiction.

(ii) \( ((P \vee q) \wedge \neg p) \rightarrow q \)
The entries in the last column are all True.
\( \implies \) The given statement is a Tautology.

(iii) \( (p \rightarrow q) \rightarrow (\neg p \rightarrow q) \)
The entries in the last column are a combination of True and False.
\( \implies \) The given statement is a contingency.

(iv) \( ((p \rightarrow q) \wedge (q \rightarrow r)) \rightarrow (p \rightarrow r) \)
All the entries in the last column are True.
\( \implies \) The given statement is a tautology. This method of using truth tables helps to classify logical statements based on their truth behavior.
In simple words: We check the final column of each truth table. If all values are True, it's a "tautology". If all are False, it's a "contradiction". If it has both True and False values, it's a "contingency".

🎯 Exam Tip: A tautology is always true, a contradiction is always false, and a contingency is sometimes true and sometimes false. The final column of the truth table determines the classification.

 

Question 8. Show that (i) \( \neg (p \wedge q) \equiv \neg P \vee \neg q \) (ii) \( \neg (p \rightarrow q) \equiv p \wedge \neg q \)
Answer:
(i) Show that \( \neg (p \wedge q) \equiv \neg P \vee \neg q \)

The entries in the columns corresponding to \( \neg (p \wedge q) \) and \( \neg P \vee \neg q \) are identical.
\( \implies \) Hence they are equivalent.

(ii) Show that \( \neg (p \rightarrow q) \equiv p \wedge \neg q \)
The entries in the columns corresponding to \( \neg (p \rightarrow q) \) and \( p \wedge \neg q \) are identical.
\( \implies \) Hence they are equivalent. These demonstrations using truth tables confirm de Morgan's laws and other important logical equivalences.
In simple words: We make truth tables for both sides of the "equals" sign ( \( \equiv \) ). If the final columns are exactly the same, it means the two statements are logically equivalent, meaning they always have the same truth value.

🎯 Exam Tip: To prove logical equivalence using truth tables, create a column for each expression and compare their final truth values. If the columns are identical, the statements are equivalent.

 

Question 9. Prove that \( q \rightarrow p \equiv \neg p \rightarrow \neg q \)
Answer:

The entries in the columns corresponding to \( q \rightarrow p \) and \( \neg P \rightarrow \neg q \) are identical.
\( \implies \) Hence they are equivalent.
\( \therefore q \rightarrow p \equiv \neg P \rightarrow \neg q \)
Hence proved. This shows that the original implication and its contrapositive are logically equivalent, a fundamental rule in logic.
In simple words: We build a truth table for \( q \rightarrow p \) and another for \( \neg p \rightarrow \neg q \). We see that their final columns are exactly the same. This proves that they are logically the same, meaning if one is true, the other is true, and if one is false, the other is false.

🎯 Exam Tip: This equivalence is a key principle: an implication is always logically equivalent to its contrapositive. Proving this using truth tables is a common exercise.

 

Question 10. Show that \( p \rightarrow q \) and \( q \rightarrow p \) are not equivalent
Answer:

From the table, it is clear that the columns for \( p \rightarrow q \) and \( q \rightarrow p \) are not identical.
\( \implies \) Therefore, \( p \rightarrow q \not\equiv q \rightarrow p \). This highlights an important difference between an implication and its converse.
In simple words: We compare the truth table for "If P, then Q" with the table for "If Q, then P". Since their results are not the same in all rows (especially when one is true and the other is false), they are not equivalent.

🎯 Exam Tip: The implication \( p \rightarrow q \) is not logically equivalent to its converse \( q \rightarrow p \). Remember that the order matters in conditional statements.

 

Question 11. Show that \( \neg (p \leftrightarrow q) \equiv p \leftrightarrow \neg q \)
Answer:

From the table, it is clear that the columns for \( \neg (p \leftrightarrow q) \) and \( p \leftrightarrow \neg q \) are identical.
\( \implies \) Therefore, \( \neg (p \leftrightarrow q) \equiv p \leftrightarrow \neg q \). This equivalence is useful in simplifying complex logical expressions involving biconditionals.
In simple words: We create truth tables for \( \neg (p \leftrightarrow q) \) and for \( p \leftrightarrow \neg q \). We see that the final column of both tables is identical. This proves that these two logical statements mean the same thing.

🎯 Exam Tip: The negation of a biconditional \( \neg (p \leftrightarrow q) \) is equivalent to saying that \( p \) is true if and only if \( q \) is false, or vice-versa. This is sometimes also equivalent to \( \neg p \leftrightarrow q \). Make sure to prove it with a truth table.

 

Question 12. Check whether the statement \( p \rightarrow (q \rightarrow p) \) is a tautology or a contradiction without using the truth table.
Answer:
We need to check the statement \( p \rightarrow (q \rightarrow p) \).
Using logical equivalences:
\( p \rightarrow (q \rightarrow p) \)
\( \equiv p \rightarrow (\neg q \vee p) \) [by implication law: \( A \rightarrow B \equiv \neg A \vee B \)]
\( \equiv \neg p \vee (\neg q \vee p) \) [by implication law]
\( \equiv \neg p \vee (p \vee \neg q) \) [by commutative law: \( A \vee B \equiv B \vee A \)]
\( \equiv (\neg p \vee p) \vee \neg q \) [by associative law: \( (A \vee B) \vee C \equiv A \vee (B \vee C) \)]
\( \equiv T \vee \neg q \) [Since \( \neg p \vee p \) is always True (T), by Tautology law]
\( \equiv T \) [Since \( T \vee A \) is always True (T), by Tautology law]
\( \implies \) Hence, \( p \rightarrow (q \rightarrow p) \) is a Tautology. This shows that the statement is always true, regardless of the truth values of p and q.
In simple words: We change the logical statement step-by-step using known rules (like changing "if-then" to "not...or..."). We keep simplifying until we get "True". If it simplifies to "True" always, it's a tautology.

🎯 Exam Tip: When proving tautologies or contradictions without truth tables, use logical equivalence laws (e.g., implication, De Morgan's, commutative, associative, distributive, identity, negation laws) to simplify the expression until you reach 'T' (for tautology) or 'F' (for contradiction).

 

Question 13. Using the truth table check whether the statements \( \neg (p \vee q) \vee (\neg p \wedge q) \) and \( \neg P \) are logically equivalent.
Answer:

From the table, it is clear that \( \neg (p \vee q) \vee (\neg p \wedge q) \) and \( \neg P \) have identical entries in their columns.
\( \implies \) Therefore, \( \neg (p \vee q) \vee (\neg p \wedge q) \equiv \neg P \). This demonstrates how complex logical statements can simplify to simpler forms.
In simple words: We create truth tables for both the long statement \( \neg (p \vee q) \vee (\neg p \wedge q) \) and the short statement \( \neg P \). We then look at their final columns. Since these columns are exactly the same, it means the two statements are logically equivalent.

🎯 Exam Tip: To check for logical equivalence between two statements using truth tables, ensure that their final columns have the exact same truth values for every row (every possible combination of inputs).

 

Question 14. Prove \( p \rightarrow (q \rightarrow r) \equiv (p \wedge q) \rightarrow r \) without using the truth table.
Answer:
We need to prove \( p \rightarrow (q \rightarrow r) \equiv (p \wedge q) \rightarrow r \).
Let's start with the left-hand side (LHS):
LHS \( = p \rightarrow (q \rightarrow r) \)
\( \equiv p \rightarrow (\neg q \vee r) \) [by implication law: \( A \rightarrow B \equiv \neg A \vee B \)]
\( \equiv \neg p \vee (\neg q \vee r) \) [by implication law]
\( \equiv (\neg p \vee \neg q) \vee r \) [by associative law for \( \vee \)]
\( \equiv \neg (p \wedge q) \vee r \) [by De Morgan's law: \( \neg A \vee \neg B \equiv \neg (A \wedge B) \)]
\( \equiv (p \wedge q) \rightarrow r \) [by implication law: \( \neg A \vee B \equiv A \rightarrow B \)]
\( \implies \) Thus, LHS \( \equiv \) RHS.
Hence Proved. This equivalence is known as Exportation Law, which is fundamental in propositional logic for restructuring implications.
In simple words: We start with the left side of the "equals" sign and change it step-by-step using logical rules (like changing "if-then" to "not...or..." or using De Morgan's law). We keep changing it until it looks exactly like the right side. This shows they are the same.

🎯 Exam Tip: For proofs without truth tables, begin with one side of the equivalence and apply known logical laws systematically. Clearly state the law used at each step. Aim to transform one side into the exact form of the other side.

 

Question 15. Prove that \( p \rightarrow (\neg q \vee r) \equiv \neg p \vee (\neg q \vee r) \) using truth table.
Answer:

From the table, it is clear that the column of \( P \rightarrow (\neg q \vee r) \) and \( \neg P \vee (\neg q \vee r) \) are identical.
\( \implies \) Therefore, \( P \rightarrow (\neg q \vee r) \equiv \neg P \vee (\neg q \vee r) \). This confirms the implication law for the given complex expression.
In simple words: We make a big truth table with all possible true/false combinations for P, Q, and R. We then fill in columns for \( P \rightarrow (\neg q \vee r) \) and \( \neg P \vee (\neg q \vee r) \). Because the final columns for both expressions are exactly the same, it means they are logically equivalent.

🎯 Exam Tip: This equivalence demonstrates the implication law \( A \rightarrow B \equiv \neg A \vee B \) applied to a more complex 'B' term. Breaking down the complex term into its own column (\( \neg q \vee r \)) can simplify the process.