Algebra of Statements

$p\Rightarrow q$ can also be written as

Dual of $~\left(~x\vee ~y\right)=x\wedge y$ is

$~\left(p\vee q\right)\vee (~p\wedge q)$ is logically equivalent to?

Solve the following problems related to three statements $\mathrm{p},\mathrm{q}$ and $\mathrm{r}$ :

If the truth value of $\left(~\mathrm{p}\wedge \mathrm{q}\right)\wedge \mathrm{r}$ is $1$ state the truth value of $(\mathrm{p}\vee \mathrm{q})\wedge \mathrm{r}$.

Solve the following problems related to three statements $\mathrm{p},\mathrm{q}$ and $\mathrm{r}$ :

If the truth value of $\mathrm{p}\vee \left(~\mathrm{p}\wedge \mathrm{q}\right)$ is $0$ or F, determine the truth value of $\mathrm{q}$.

Solve the following problems related to three statements $\mathrm{p},\mathrm{q}$ and $\mathrm{r}$ :

If $\mathrm{p}\wedge \left(~\mathrm{p}\vee \mathrm{q}\right)$ takes the truth value T, find the truth values of $\mathrm{p}, \mathrm{q}$ and $\mathrm{p}\wedge \mathrm{q}$.

For two statements $\mathrm{p}$ and $\mathrm{q}$, prove the following relations of logical equivalence.

$\mathrm{p}\to \mathrm{p}\wedge \mathrm{q}\equiv \mathrm{p}\to \mathrm{q}$

For two statements $\mathrm{p}$ and $\mathrm{q}$, prove the following relations of logical equivalence.

$\mathrm{p}\vee \mathrm{q}\to \mathrm{p}\equiv \mathrm{p}\vee ~\mathrm{q}$

For two statements $\mathrm{p}$ and $\mathrm{q}$, prove the following relations of logical equivalence.

$\left(\mathrm{p}\to \mathrm{q}\right)\to \mathrm{q}\equiv \mathrm{p}\vee \mathrm{q}$

For two statements $\mathrm{p}$ and $\mathrm{q}$, prove the following relations of logical equivalence.

$\mathrm{p}\wedge \left(~\mathrm{p}\vee \mathrm{q}\right)\equiv \mathrm{p}\wedge \mathrm{q}$

If $\mathrm{A}$ is a tautology, then for any statement $\mathrm{p}$, show that $\mathrm{A}\wedge \mathrm{p}\equiv \mathrm{p}$.

If $p$ and $q$ are two logical statements, then which of the following is not commutative ?

Let two statements $p$ and $q$ is defined as below

$p : 2\times 4=7$,

$q :$ Kota is in London.

Then truth values of these statements: $\left(i\right) p\leftrightarrow q$             $\left(ii\right) ~p\to q$              $\left(iii\right) \left(p\to q\right)\wedge q$ are

The dual of $p\wedge ~p\equiv F$ is

$p\vee (p\wedge q)$ is equivalent to -

The logical equivalent statement of $p\leftrightarrow q$ is

The proposition $~(p\vee q)\vee (~p\wedge q)$ is logically equivalent to

$\left(~\left(p\vee q\right)\right)\vee \left(~p\wedge q\right)$ is logically equivalent to

$\sim \left(p\vee q\right)\vee \left(\sim p\wedge q\right)$ is equivalent to

The statement $~p\to \left(q\to p\right)$ is equivalent to