JEE Main & Advanced Mathematics Mathematical Logic and Boolean Algebra Algebra of Statements

In the previous section, we have seen that statements satisfy many standard results. In this section, we shall state those results as laws of algebra of statements.

The following are some laws of algebra of statements.

(i) Idempotent laws : For any statement p, we have

(a) \[p\vee p\equiv p\]                                

(b) \[p\wedge p\equiv p\]

(ii) Commutative laws : For any two statements p and q, we have

(a) \[p\vee q\equiv q\vee p\]                   

(b) \[p\wedge p\equiv q\wedge p\]

(iii) Association laws : For any three statements \[p,q,r\]  we have

(a) \[(p\vee q)\vee r\equiv p\vee (q\vee r)\]   

(b) \[(p\wedge q)\wedge r\equiv p\wedge (q\wedge r)\]

(iv) Distributive laws : For any three statements \[p,q,r\] we have

(a) \[p\wedge (p\vee q)\equiv (p\wedge q)\vee (q\wedge r)\]

(b) \[p\vee (p\wedge q)\equiv (p\vee q)\wedge (q\vee r)\]

(v) Demorgan’s laws : If p and q are two statements, then

(a) \[\tilde{\ }(p\wedge q)\equiv \tilde{\ }p\vee \tilde{\ }q\]     

(b) \[\tilde{\ }(p\vee q)\equiv \tilde{\ }p\wedge \tilde{\ }q\]

(vi) Identity laws : If t and c denote a tautology and a contradiction respectively, then for any statement p, we have

(a) \[p\wedge t\equiv p\]           

(b) \[p\vee c\equiv p\]

(c) \[p\vee t\equiv t\] 

(d) \[p\wedge c\equiv c\]

(vii) Complement laws : For any statements p, we have

(a) \[p\vee \tilde{\ }p=t\]

(b) \[p\wedge \tilde{\ }p=c\]

(c) \[\tilde{\ }t=c\]  

(d) \[\tilde{\ }c=t\]

where t and c denote a tautology and a contradiction respectively.

(viii) Law of contrapositive : For any two statements p and q, we have

\[p\Rightarrow q\equiv \,\tilde{\ }q\Rightarrow \,\tilde{\ }p\]

(ix) Involution laws : For any statement p, we have \[\tilde{\ }(\tilde{\ }p)\equiv p\]