By using truth table prove that, for all statements P and Q, the three statement

Question

By using truth table prove that, for all statements P and Q, the three statements ( i ) 'P Q', ( ii ) ' ( P or Q ) Q' and ( iii ) ' ( P and Q ) P' are equivalent. Prove that the three basic connectives 'or', 'and' and 'not' can all be written in terms of the single connective 'notand' where 'P notand Q' is interpreted as 'not ( P and Q ) '. Prove the following statements concerning positive integers a, b and c. ( a s b ) and ( a s c ) a s ( b + c ) . ( a s b ) or ( a s c ) a s bc. Which of the following conditions are necessary for the positive integer n to be divisible by 6 ( proofs not necessary ) ? 3 s n. 9 s n. 12 s n.

Explanation / Answer

GIVEN A DIVIDES B AND A DIVIDES C TPT A DIVIDES [B+C] PROOF GIVEN A DIVIDES B ...SO B=A*P....................1 WHERE P IS AN INTEGER GIVEN A DIVIDES C ...SO C=A*Q....................2 GIVEN A DIVIDES C ...SO C=A*Q....................2 WHERE Q IS AN INTEGER EQN.1+EQN.2 GIVES [B+C]=A*P+A*Q =A[P+Q]=A*INTEGER HENCE A DIVIDES [B+C] ......PROVED =================================== GIVEN A DIVIDES B OR A DIVIDES C TPT A DIVIDES B*C PROOF CASE 1 GIVEN A DIVIDES B ...SO B=A*P....................1 WHERE P IS AN INTEGER B*C=A*P*C=A*INTEGER HENCE A DIVIDES B*C ....PROVED CASE 2 GIVEN A DIVIDES C ...SO C=A*Q....................2 CASE 1 GIVEN A DIVIDES B ...SO B=A*P....................1 WHERE P IS AN INTEGER B*C=A*P*C=A*INTEGER HENCE A DIVIDES B*C ....PROVED CASE 2 GIVEN A DIVIDES C ...SO C=A*Q....................2 GIVEN A DIVIDES C ...SO C=A*Q....................2 WHERE Q IS AN INTEGER   B*C=B*A*Q=A*INTEGER HENCE A DIVIDES B*C ....PROVED HENCE A DIVIDES B*C ....PROVED

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