Question Number 203468 by MrGHK last updated on 19/Jan/24
$$\:\boldsymbol{\mathrm{let}}\:\boldsymbol{\mathrm{A}}\:{B}\:{and}\:{C}\:\:\boldsymbol{\mathrm{be}}\:\boldsymbol{\mathrm{sets}}\:\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{using}}\:\boldsymbol{\mathrm{venn}}\:\boldsymbol{\mathrm{diagrams}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{by}}\:\boldsymbol{\mathrm{first}}\:\boldsymbol{\mathrm{principle}} \\ $$$$\left.\:\boldsymbol{\mathrm{a}}\right)\boldsymbol{\mathrm{A}}×\left(\boldsymbol{\mathrm{B}}\cup\boldsymbol{\mathrm{C}}\right)=\left(\boldsymbol{\mathrm{A}}×\boldsymbol{\mathrm{B}}\right)\cup\left(\boldsymbol{\mathrm{A}}×\boldsymbol{\mathrm{C}}\right) \\ $$$$\left.\boldsymbol{\mathrm{b}}\right)\mathrm{A}×\left(\boldsymbol{\mathrm{B}}\cap\boldsymbol{\mathrm{C}}\right)=\left(\boldsymbol{\mathrm{A}}×\boldsymbol{\mathrm{B}}\right)\cap\left(\boldsymbol{\mathrm{A}}×\boldsymbol{\mathrm{C}}\right) \\ $$$$\left.\boldsymbol{\mathrm{c}}\right)\left(\boldsymbol{\mathrm{A}}−\boldsymbol{\mathrm{B}}\right)\cap\left(\boldsymbol{{A}}\cap\boldsymbol{\mathrm{B}}\right)\cap\left(\boldsymbol{\mathrm{B}}−\boldsymbol{\mathrm{A}}\right)=\left\{\right\} \\ $$