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प्रश्न
Write the converse, inverse, and contrapositive of the following statement.
If he studies, then he will go to college.
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उत्तर
Let p: He studies.
q: He will go to college.
∴ The given statement is p → q.
Its converse is q → p.
If he will go to college then he studies.
Its inverse is ~ p → ~ q.
If he does not study then he will not go to college.
Its contrapositive is ~ q → ~ p.
If he will not go to college then he does not study.
संबंधित प्रश्न
If p : It is raining
q : It is humid
Write the following statements in symbolic form:
(a) It is raining or humid.
(b) If it is raining then it is humid.
(c) It is raining but not humid.
Using truth table, examine whether the following statement pattern is tautology, contradiction or contingency: p ∨ [∼(p ∧ q)]
Use the quantifiers to convert the following open sentence defined on N into true statement:
x2 ≥ 1
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 ∧ q) (p → r)
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 → r)] ↔ [(p ∧ q) → r]
Prove that the following statement pattern is a tautology.
(p → q) ↔ (~ q → ~ p)
Prove that the following statement pattern is a contradiction.
(p ∨ q) ∧ (~p ∧ ~q)
Prove that the following statement pattern is a contradiction.
(p ∧ q) ∧ (~p ∨ ~q)
Prove that the following statement pattern is a contradiction.
(p → q) ∧ (p ∧ ~ q)
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 → (p → q) ≡ ~ q → (p → q)
Write the dual of the following:
(p ∨ q) ∨ r
Write the dual of the following:
~(p ∨ q) ∧ [p ∨ ~ (q ∧ ~ r)]
Write the dual statement of the following compound statement.
A number is a real number and the square of the number is non-negative.
Write the negation of the following statement.
∀ n ∈ N, n + 1 > 0
Write the converse, inverse, and contrapositive of the following statement.
"If it snows, then they do not drive the car"
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) → ~(q ∧ r)
Using the truth table, prove the following logical equivalence.
p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
State the dual of the following statement by applying the principle of duality.
(p ∧ ~q) ∨ (~ p ∧ q) ≡ (p ∨ q) ∧ ~(p ∧ q)
Write the dual of the following
(p ˄ ∼q) ˅ (∼p ˄ q) ≡ (p ˅ q) ˄ ∼(p ˄ q)
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`
Which of the following is not true for any two statements p and q?
Show that the following statement pattern is a contingency:
(p→q)∧(p→r)
The converse of contrapositive of ∼p → q is ______.
