Introduction of Mathematical Logic

Determine the truth values of $p$ and $q$ in the following cases:

$\left(p\wedge q\right)$ is $F$ and $\left(p\wedge q\right)\to q$ is $T$

Determine the truth values of $p$ and $q$ in the following cases:

$\left(p\vee q\right)$ is $T$ and $\left(p\vee q\right)\to q$ is $F$

Find truth values of $p$and$q$ in the following cases:

$\left(p\vee q\right)$ is $T$ and $\left(p\wedge q\right)$ is $T$.

Write the truth values of the following statements:

In $∆ABC$ if all sides are equal then its all angles are equal.

Write the truth values of the following statements:

The square of any even number is odd or the cube of any odd number is odd.

Write the truth values of the following statements:

$\sqrt{5}$ is an irrational but $3\sqrt{5}$ is a complex number.

State which of the following are statements. Justify. In case of statement, state its truth value.

If $x$ is a whole number then $x+6=0$

Which of the following sentences are statements in logic? Justify. Write down the truth value of the statements:

${\cos }^2\theta -{\sin }^2\theta =\cos 2\theta \text{for all }\theta \in R.$

Which of the following sentences are statements in logic? Justify. Write down the truth value of the statements:

Please get me a glass of water.

Which of the following sentences are statements in logic? Justify. Write down the truth value of the statements:

India is a country and Himalayas is a river.

Which of the following sentences are statements in logic? Justify. Write down the truth value of the statements:

$\pi$ is an irrational number.

Write the negation of the following.

$7+3>5$

Write the negation of the following.

Polygon $ABCDE$ is a pentagon.

Write the negation of the following.

$\sqrt{2}$ is a rational number.

Write the negation of the following.

$3$ is not a root of the equation $x^2+3x-18=0$

If the statements $p,q$ are true statements and $r,s$ are false statements then determine the truth values of the following.

$~\left[\left(~p\wedge r\right)\vee \left(s\to ~q\right)\right]\leftrightarrow \left(p\wedge r\right)$

If the statements $p,q$ are true statements and $r,s$ are false statements then determine the truth values of the following.

$\left[\left(~p\wedge q\right)\wedge ~r\right]\vee \left[\left(q\to p\right)\to \left(~s\vee r\right)\right]$

If the statements $p,q$ are true statements and $r,s$ are false statements then determine the truth values of the following.

$\left[~p\wedge \left(~q\wedge r\right)\right]\vee \left[\left(q\wedge r\right)\vee \left(p\wedge r\right)\right]$

If the statements $p,q$ are true statements and $r,s$ are false statements then determine the truth values of the following.

$\left(~r\leftrightarrow p\right)\to ~q$

If the statements $p,q$ are true statements and $r,s$ are false statements then determine the truth values of the following.

$\left(p\to q\right)\wedge ~r$