Question
Construct the truth table for each of the following statement patterns.
i) $\quad p \rightarrow(q \rightarrow p)$
i) $\quad p \rightarrow(q \rightarrow p)$
✓
Answer
| p | q | q $\rightarrow$ p | p $\rightarrow$(q $\rightarrow$ p) |
| T | T | T | T |
| T | F | T | T |
| F | T | F | T |
| F | F | T | T |
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Similar questions
If $\bar{a}=3 \hat{i}+4 \hat{j}-5 \hat{k}$ and $\bar{b}=3 \hat{i}-4 \hat{j}-5 \hat{k}$
(i) find $\bar{a} \cdot \bar{b}$
(ii) the angle between $\bar{a}$ and $\bar{b}$.
(iii) the scalar projection of $\bar{a}$ in the direction of $\bar{b}$.
(iv) the vector projection of $\bar{b}$ along $\bar{a}$.
→(i) find $\bar{a} \cdot \bar{b}$
(ii) the angle between $\bar{a}$ and $\bar{b}$.
(iii) the scalar projection of $\bar{a}$ in the direction of $\bar{b}$.
(iv) the vector projection of $\bar{b}$ along $\bar{a}$.
Which of the following sentences are statements in logic ? Write down the truth values of
the statements.
i) $6 × 4 = 25$
ii) $x + 6 = 9$
iii) What are you doing ?
iv) The quadratic equation $x^2 – 5x + 6 = 0$ has $2$ real roots.
v) Please, sit down
vi) The Moon revolves around the earth.
vii) Every real number is a complex number.
viii) He is honest.
ix) The square of a prime number is a prime number.
→the statements.
i) $6 × 4 = 25$
ii) $x + 6 = 9$
iii) What are you doing ?
iv) The quadratic equation $x^2 – 5x + 6 = 0$ has $2$ real roots.
v) Please, sit down
vi) The Moon revolves around the earth.
vii) Every real number is a complex number.
viii) He is honest.
ix) The square of a prime number is a prime number.
Express the following compound statements symbolically without examining the truth values.
i) 2 is an even number and 25 is a perfect square.
ii) A school is open or there is a holiday.
iii) Delhi is in India but Dhaka is not in Srilanks.
iv) 3 + 8 ≥ 12 if and only if 5 × 4 ≤ 25.
i) 2 is an even number and 25 is a perfect square.
ii) A school is open or there is a holiday.
iii) Delhi is in India but Dhaka is not in Srilanks.
iv) 3 + 8 ≥ 12 if and only if 5 × 4 ≤ 25.
Construct the truth table for each of the following:
(iii) ~(~p ∧ ~q) ∨ q
(iii) ~(~p ∧ ~q) ∨ q
Find the principal values of the following :
(i) $\sin ^{-1}\left(-\frac{1}{2}\right)$
(ii) $\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)$
(iii) $\cot ^{-1}\left(-\frac{1}{\sqrt{3}}\right)$
→(i) $\sin ^{-1}\left(-\frac{1}{2}\right)$
(ii) $\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)$
(iii) $\cot ^{-1}\left(-\frac{1}{\sqrt{3}}\right)$
If $A=\{1,2,3,4,5,6,7\}$, determine the truth value of the following.
i) $\exists x \in A$ such that $x-4=3$
ii) $\forall x \in A , \quad x+1 \geq 3$
iii) $\forall x \in A , 8-x \leq 7$
iv) $\exists x \in A$, such that $x+8=16$
→i) $\exists x \in A$ such that $x-4=3$
ii) $\forall x \in A , \quad x+1 \geq 3$
iii) $\forall x \in A , 8-x \leq 7$
iv) $\exists x \in A$, such that $x+8=16$
Prove the following
(i) $2 \tan ^{-1}\left(-\frac{1}{3}\right)+\cos ^{-1}\left(\frac{3}{5}\right)=\frac{\pi}{2}$
(ii) $2 \tan ^{-1}\left(\frac{1}{3}\right)+\tan ^{-1}\left(\frac{1}{7}\right)=\frac{\pi}{4}$
→(i) $2 \tan ^{-1}\left(-\frac{1}{3}\right)+\cos ^{-1}\left(\frac{3}{5}\right)=\frac{\pi}{2}$
(ii) $2 \tan ^{-1}\left(\frac{1}{3}\right)+\tan ^{-1}\left(\frac{1}{7}\right)=\frac{\pi}{4}$
State the vectors which are :
(i) equal in magnitude
(ii) parallel
(iii) in the same direction
(iv) equal
(v) negatives of one another
(i) equal in magnitude
(ii) parallel
(iii) in the same direction
(iv) equal
(v) negatives of one another
Write the negations of the following.
i) Price increases
ii) 0! ≠ 1
iii) 5 + 4 = 9
i) Price increases
ii) 0! ≠ 1
iii) 5 + 4 = 9
Find the values of the following :(i) $\sin \left[\sin ^{-1}\left(\frac{3}{5}\right)+\cos ^{-1}\left(\frac{3}{5}\right)\right]$
(ii) $\cos \left[\cos ^{-1}\left(-\frac{1}{2}\right)+\tan ^{-1} \sqrt{3}\right]$
→(ii) $\cos \left[\cos ^{-1}\left(-\frac{1}{2}\right)+\tan ^{-1} \sqrt{3}\right]$