Algebra of Statements

Select the associative law of statements from the following.

What is the identity law with "and"?

For any statement $a$.

Draw the simplified switching circuit for the following switching circuit.

Give an alternate arrangement for the circuit, so that the new circuit has two switches only.

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

The dual of $p\wedge ~p\equiv F$ is

In the above network, current does not flow, when

If $p: 25$ is a factor of $625$ and $q: 169$ is a perfect square, then $~\left(p\Rightarrow q\right)$ is equivalent to

Which of the following connectives satisfy commutative law?

Which of the following pairs are logically equivalent?

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

Dual of ${\left(x^{\prime }\wedge y^{\prime }\right)}^{\prime }=x\vee y$ is

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

Simplify the following circuit and find the boolean polynomial.