Prove the following in 2 ways a) by contraposition and b) by contradiction 26. F

Question

Prove the following in 2 ways
a) by contraposition and b) by contradiction 26. For all integers a, b, and c, if a |b  and ~(a | c) then ~(a | (b+c)) (If a divides b and a does not divide c, then a does notdivide (b + c) ) Hint: to prove p-> q V r, it suffices to prove either p^~p-> r or p ^ ~r -> q. Prove the following in 2 ways
a) by contraposition and b) by contradiction 26. For all integers a, b, and c, if a |b  and ~(a | c) then ~(a | (b+c)) (If a divides b and a does not divide c, then a does notdivide (b + c) ) Hint: to prove p-> q V r, it suffices to prove either p^~p-> r or p ^ ~r -> q.

Explanation / Answer

QuestionDetails: Prove the following in 2 ways
a) by contraposition and b) by contradiction 26. For all integers a, b, and c, if a |b  and ~(a | c) then ~(a | (b+c)) (If a divides b and a does not divide c, then a does notdivide (b + c) ) Hint: to prove p-> q V r, it suffices to prove either p^~p-> r or p ^ ~r -> q.
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a) by contraposition
THAT IS
A|[B+C] DOES NOT IMPLY THAT A|B AND ˜A|C
THAT IS GIVEN
A|[B+C] AND A|C
TST A|B
A|[B+C]
B+C=A*P.................1
WHERE P IS AN INTEGER
A|C
C=A*Q.....................2
WHERE P IS AN INTEGER
PUTTING EQN.2 IN EQN.1
B+A*Q=A*P
B=A*[P-Q]=A*INTEGER...
THAT IS A DIVIDES B ....PROVED.
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b) by contradiction
PROOF BY CONTRADICTION...
GIVEN
A|B.......
THAT IS
B=A*P .....................1
WHERE P IS AN INTEGER.
A DOES NOT DIVIDE C...............2
TPT
A DOES NOT DIVIDE B+C.......
LET US ASSUME
A|(B+C)............
THAT IS
B+C=A*Q..................2
WHERE Q IS AN INTEGER
SUBSTITUTING FOR B FROM EQN.1
A*P+C=A*Q
C=A*[Q-P] = C*INTEGER SINCE Q AND P BEING INTEGERS Q-P IS ANINTEGER.
THAT IS A DIVIDES C
BUT THIS IS CONTRARY TO THE GIVEN STATEMENT THAT A DOES DIVIDEC
SO OUR ASSUMPTION THAT A|[B+C] IS NOT CORRECT
THAT IS ..A DOES NOT DIVIDE [B+C].....PROVED

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