Advertisements
Advertisements
Question
Using the truth table proves the following logical equivalence.
∼ (p ↔ q) ≡ (p ∧ ∼ q) ∨ (q ∧ ∼ p)
Solution
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| p | q | ∼ p | ∼ q | p ↔ q | ∼ (p ↔ q) | p ∧ ∼ q | q ∧ ∼ p | (p ∧ ∼ q) ∨ (q ∧ ∼ p) |
| T | T | F | F | T | F | F | F | F |
| T | F | F | T | F | T | T | F | T |
| F | T | T | F | F | T | F | T | T |
| F | F | T | T | T | F | F | F | F |
The entries in columns 6 and 9 are identical.
∴ ∼ (p ↔ q) ≡ (p ∧ ∼ q) ∨ (q ∧ ∼ p)
APPEARS IN
RELATED QUESTIONS
Prove that the following statement pattern is equivalent :
(p ∨ q) → r and (p → r) ∧ (q → r)
Write converse and inverse of the following statement:
“If a man is a bachelor then he is unhappy.”
Prove that the following statement pattern is a tautology : ( q → p ) v ( p → q )
If p and q are true statements and r and s are false statements, find the truth value of the following :
( p ∧ ∼ r ) ∧ ( ∼ q ∧ s )
Using truth table, examine whether the following statement pattern is tautology, contradiction or contingency: p ∨ [∼(p ∧ q)]
Show that the following statement pattern in contingency :
(~p v q) → [p ∧ (q v ~ q)]
Use the quantifiers to convert the following open sentence defined on N into true statement:
x2 ≥ 1
Prove that the following statement pattern is equivalent:
(p v q) → r and (p → r) ∧ (q → r)
Write the negation of the Following Statement :
∀ y ∈ N, y2 + 3 ≤ 7
Write the negation of the following statement :
If the lines are parallel then their slopes are equal.
State if the following sentence is a statement. In case of a statement, write down the truth value :
Every quadratic equation has only real roots.
By constructing the truth table, determine whether the following statement pattern ls a tautology , contradiction or . contingency. (p → q) ∧ (p ∧ ~ q ).
Using the truth table prove the following logical equivalence.
p ↔ q ≡ ∼ [(p ∨ q) ∧ ∼ (p ∧ q)]
Using the truth table prove the following logical equivalence.
(p ∨ q) → r ≡ (p → r) ∧ (q → r)
Using the truth table prove the following logical equivalence.
p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
Using the truth table prove the following logical equivalence.
p → (q ∧ r) ≡ (p ∧ q) (p → r)
Using the truth table prove the following logical equivalence.
[∼ (p ∨ q) ∨ (p ∨ q)] ∧ r ≡ r
Examine whether the following statement pattern is a tautology or a contradiction or a contingency.
[(p → q) ∧ ∼ q] → ∼ p
Examine whether the following statement pattern is a tautology or a contradiction or a contingency.
(p ↔ q) ∧ (p → ∼ q)
Determine whether the following statement pattern is a tautology, contradiction, or contingency:
(p → q) ∧ (p ∧ ∼q)
Determine whether the following statement pattern is a tautology, contradiction or contingency:
(p ∧ q) ∨ (∼p ∧ q) ∨ (p ∨ ∼q) ∨ (∼p ∧ ∼q)
Determine whether the following statement pattern is a tautology, contradiction or contingency:
[(p ∨ ∼q) ∨ (∼p ∧ q)] ∧ r
Prepare truth tables for the following statement pattern.
(~ p ∨ q) ∧ (~ p ∨ ~ q)
Prove that the following statement pattern is a tautology.
(p → q) ↔ (~ q → ~ p)
Prove that the following statement pattern is a tautology.
(~ p ∨ ~ q) ↔ ~ (p ∧ q)
If p is any statement then (p ∨ ∼p) is a ______.
Show that the following statement pattern is contingency.
(p∧~q) → (~p∧~q)
Show that the following statement pattern is contingency.
p ∧ [(p → ~ q) → q]
Using the truth table, verify
~(p ∨ q) ∨ (~ p ∧ q) ≡ ~ p
Prove that the following pair of statement pattern is equivalent.
p ↔ q and (p → q) ∧ (q → p)
Write the dual of the following:
(p ∨ q) ∨ r
Write the dual statement of the following compound statement.
13 is prime number and India is a democratic country.
Write the dual statement of the following compound statement.
A number is a real number and the square of the number is non-negative.
Using the rules of negation, write the negation of the following:
(p → r) ∧ q
Using the rules of negation, write the negation of the following:
~(p ∨ q) → r
Using the rules of negation, write the negation of the following:
(~p ∧ q) ∧ (~q ∨ ~r)
With proper justification, state the negation of the following.
(p ↔ q) v (~ q → ~ r)
With proper justification, state the negation of the following.
(p → q) ∧ r
Construct the truth table for the following statement pattern.
(p ∧ r) → (p ∨ ~q)
What is tautology? What is contradiction?
Show that the negation of a tautology is a contradiction and the negation of a contradiction is a tautology.
Determine whether the following statement pattern is a tautology, contradiction, or contingency.
[(p ∧ q) ∨ (~p)] ∨ [p ∧ (~ q)]
Determine whether the following statement pattern is a tautology, contradiction, or contingency.
[(~p ∧ q) ∧ (q ∧ r)] ∨ (~q)
Determine whether the following statement pattern is a tautology, contradiction, or contingency.
[~(p ∨ q) → p] ↔ [(~p) ∧ (~q)]
Using the truth table, prove the following logical equivalence.
p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
Using the truth table, prove the following logical equivalence.
p ∧ (~p ∨ q) ≡ p ∧ q
Using the truth table, prove the following logical equivalence.
p ↔ q ≡ ~(p ∧ ~q) ∧ ~(q ∧ ~p)
Write the converse, inverse, contrapositive of the following statement.
If I do not work hard, then I do not prosper.
State the dual of the following statement by applying the principle of duality.
(p ∧ ~q) ∨ (~ p ∧ q) ≡ (p ∨ q) ∧ ~(p ∧ q)
State the dual of the following statement by applying the principle of duality.
2 is even number or 9 is a perfect square.
Write the dual of the following.
(~p ∧ q) ∨ (p ∧ ~q) ∨ (~p ∧ ~q)
Write the dual of the following.
(p ∧ q) ∧ r ≡ p ∧ (q ∧ r)
Write the dual of the following.
p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (q ∨ r)
Express the truth of the following statement by the Venn diagram.
Some members of the present Indian cricket are not committed.
The false statement in the following is ______.
Write the dual of the following
(p ˄ ∼q) ˅ (∼p ˄ q) ≡ (p ˅ q) ˄ ∼(p ˄ q)
Choose the correct alternative:
If p → q is an implication, then the implication ~q → ~p is called its
The contrapositive of p → ~ q is ______
Complete the truth table.
| p | q | r | q → r | r → p | (q → r) ˅ (r → p) |
| T | T | T | T | `square` | T |
| T | T | F | F | `square` | `square` |
| T | F | T | T | `square` | T |
| T | F | F | T | `square` | `square` |
| F | T | T | `square` | F | T |
| F | T | F | `square` | T | `square` |
| F | F | T | `square` | F | T |
| F | F | F | `square` | T | `square` |
The given statement pattern is a `square`
If p → (∼p v q) is false, then the truth values of p and q are respectively
The statement pattern (p ∧ q) ∧ [~ r v (p ∧ q)] v (~ p ∧ q) is equivalent to ______.
Which of the following is not equivalent to p → q.
The equivalent form of the statement ~(p → ~ q) is ______.
Which of the following is not true for any two statements p and q?
Using truth table verify that:
(p ∧ q)∨ ∼ q ≡ p∨ ∼ q
Show that the following statement pattern is a contingency:
(p→q)∧(p→r)
