Question

In: Advanced Math

Prove by contraposition and again by contradiction: For all integers a,b, and c, if a divides...

Prove by contraposition and again by contradiction:

For all integers a,b, and c, if a divides b and a does not divide c then a does not divide b + c

Elaboration with definitions / properties used would be appreciated!

Thanks in advance!!

Solutions

Expert Solution

Proof by Contraposition : Suppose divides .

Then there exist such that

Now if divides then there exist such that .

is a multiple of and so divides .

So it is not true that divides and does not divides .

Hence by contraposition if divides and does not divides then does not divides .

Proof by Contradiction : Given divides and does not divides .

Suppose divides .

Then there exist such that

As divides then there exist such that .

is a multiple of and so divides .

a contradiction to does not divides .

Hence if divides and does not divides then does not divides .

Please comment if needed .

RATE THE ANSWER ACCORDINGLY .

Colby Messinger answered 1 month ago

Next > < Previous

Related Solutions

Contradiction proof conception Prove: If A is true, then B is true Contradiction: If A is...

Contradiction proof conception Prove: If A is true, then B is true Contradiction: If A is true, then B is false. so we suppose B is false and follow the step to prove. At the end we get if A is true then B is true so contradict our assumption However, Theorem: Let (xn) be a sequence in R. Let L∈R. If every subsequence of (xn) has a further subsequence that converges to L, then (xn) converges to L. Proof: Assume,...

Prove That For All Natural Numbers A > 1 And B > 1, If A Divides...

Prove That For All Natural Numbers A > 1 And B > 1, If A Divides B Then A Does Not Divide B+1 (prove by contradiction)

For each of the statements, begin a proof by contraposition and a proof by contradiction. This...

For each of the statements, begin a proof by contraposition and a proof by contradiction. This will include rewriting the statement, writing the assumptions, and writing what needs to be shown. From there, pick one of the two methods and finish the proof. a) For all integers m and n, if m + n is even the m and n are both even or m and n are both odd. b) For all integers a, b, and c, if a...

Prove or disprove: If a, b, c are any three distinct positive integers such that 1/a...

Prove or disprove: If a, b, c are any three distinct positive integers such that 1/a + 1/b + 1/c = 1, then a + b + c is a prime

Prove or disprove: If a, b, c are any three distinct positive integers such that 1/a...

Prove or disprove: If a, b, c are any three distinct positive integers such that 1/a + 1/b + 1/c = 1, then a + b + c is a prime.

Prove that 3 divides n^3 −n for all n ≥ 1.

a.) Prove the following: Lemma. Let a and b be integers. If both a and b...

a.) Prove the following: Lemma. Let a and b be integers. If both a and b have the form 4k+1 (where k is an integer), then ab also has the form 4k+1. b.)The lemma from part a generalizes two products of integers of the form 4k+1. State and prove the generalized lemma. c.) Prove that any natural number of the form 4k+3 has a prime factor of the form 4k+3.

Use boolean algebra to prove that: (A^- BC^-) + (A^- BC) + (A* B^- *C) +...

Use boolean algebra to prove that: (A^- *B*C^-) + (A^- *B*C) + (A* B^- *C) + (A*B* C^-) + (A*B*C)= (A+B)*(B+C) A^- is same as "not A" please show steps to getting the left side to equal the right side, use boolean algebra properties such as distributive, absorption,etc

Find 3 definitions of e. Prove they are equivalent (transitivity: a=b, b=c, and a=c) prove the...

Find 3 definitions of e. Prove they are equivalent (transitivity: a=b, b=c, and a=c) prove the 3 defintions of e are equivalent.

Prove the transitive property of similarity: if A~B and B~C, then A~C.

Subjects