JEE Main & Advanced Mathematics Mathematical Logic and Boolean Algebra Duality

Definition : Two compound statements \[{{S}_{1}}\] and \[{{S}_{2}}\] are said to be duals of each other if one can be obtained from the other by replacing \[\wedge \]by \[\vee \] and \[\vee \]by \[\wedge \].

• The connective \[\wedge \]and \[\vee \]are also called duals of each other

• If a compound statements contains the special variable

t (tautology) or c (contradiction),  then to obtain its dual we replace t by c and c by t in addition to replacing \[\wedge \]by \[\vee \]and \[\vee \]by \[\wedge \].

• Let \[S(p,q)\]be a compound statement containing two sub- statements and \[{{S}^{*}}\](p, q) be its dual. Then,

(i) \[\tilde{\ }S(p,q)\equiv {{S}^{*}}(\tilde{\ }p,\tilde{\ }q)\]

(ii) \[\tilde{\ }{{S}^{*}}(p,q)\equiv S(\tilde{\ }p,\tilde{\ }q)\]

• The above result can be extended to the compound statements having finite number of sub- statements. Thus, if \[S({{p}_{1}},{{p}_{2}},....{{p}_{n}})\] is a compound statement containing n sub-statement \[{{p}_{1}},{{p}_{2}},....,{{p}_{n}}\] and \[{{S}^{*}}({{p}_{1}}{{p}_{2}},....,{{p}_{n}})\] is its dual. Then,

(i) \[\tilde{\ }S({{p}_{1}},{{p}_{2}},....,{{p}_{n}})\equiv {{S}^{*}}(\tilde{\ }{{p}_{1}},\tilde{\ }{{p}_{2}},....,\tilde{\ }{{p}_{n}})\]

(ii) \[\tilde{\ }{{S}^{*}}({{p}_{1}},{{p}_{2}},....,{{p}_{n}})\equiv S(\tilde{\ }{{p}_{1}},\tilde{\ }{{p}_{2}},....,\tilde{\ }{{p}_{n}})\]