Algebra of Statements

Using the rules of negation write the negation of the statement $~p\dot{\vee }(q\to \sim r)$.

Using the rules of negation write the negation of the statement $p\wedge ~(q\wedge r)$.

Construct the new switching circuit for the following circuit with only one switch by simplifying the given circuit.

Obtain the simple logical expression for $(~p\wedge q)\vee (~p\wedge ~q)\vee (p\wedge ~q)$. Hence, draw the corresponding switching circuit.

Obtain the simple logical expression for $p\vee (q\wedge ~q)$. Hence, draw the corresponding switching circuit.

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.

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

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

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

Let \[*\] be a binary operation defined on $R$ by $a$ \[*\] $b=\frac{a+b}{4},\forall a,b\in R,$ then the operation is

Binary operation \[*\] on $R-\left\{-1\right\}$ defined by $a$\[*\]$b=\frac{a}{b+1}$ is

Which of the following connectives satisfy commutative law?

Which of the following is a contradiction?

Which of the following laws does the connective $\wedge$ (and) satisfy?

$\left(~\left(~p\right)\right)\wedge q$ is equal to

The statement $(\sim p\wedge q)\vee \sim q$ is