Fundamentals Of Proof By Cases

1) Proof by case analysis requires cases to be what?

Exhaustive and mutually exclusive

Exhaustive only

Mutually exclusive only

Optional

For a proof by case analysis to be valid, the cases must be both exhaustive and mutually exclusive. Exhaustive means that every possible scenario is covered by at least one case, ensuring that no situation is left unconsidered. Mutually exclusive means that no two cases overlap, which prevents double-counting and ensures that each scenario is handled in only one case. If cases are not exhaustive, there might be a counterexample in a missing case, invalidating the proof. If cases are not mutually exclusive, it might lead to redundancy or confusion in the proof. Therefore, both properties are essential for a sound case analysis.

Explanation

For a proof by case analysis to be valid, the cases must be both exhaustive and mutually exclusive. Exhaustive means that every possible scenario is covered by at least one case, ensuring that no situation is left unconsidered. Mutually exclusive means that no two cases overlap, which prevents double-counting and ensures that each scenario is handled in only one case. If cases are not exhaustive, there might be a counterexample in a missing case, invalidating the proof. If cases are not mutually exclusive, it might lead to redundancy or confusion in the proof. Therefore, both properties are essential for a sound case analysis.

2) To prove “P or Q implies R,” a good first step is:

Assume P and Q

Prove R directly

Use induction

Consider cases P and Q separately

When proving a statement of the form "P or Q implies R", a standard approach is to use case analysis. This involves considering two cases: one where P is true, and another where Q is true. In each case, we assume that particular hypothesis and prove that R follows. Assuming both P and Q simultaneously is incorrect because "P or Q" means at least one is true, not both. Proving R directly may not be possible without considering the cases. Induction is not directly relevant here unless dealing with natural numbers. Therefore, considering cases P and Q separately is the correct first step.

Explanation

When proving a statement of the form "P or Q implies R", a standard approach is to use case analysis. This involves considering two cases: one where P is true, and another where Q is true. In each case, we assume that particular hypothesis and prove that R follows. Assuming both P and Q simultaneously is incorrect because "P or Q" means at least one is true, not both. Proving R directly may not be possible without considering the cases. Induction is not directly relevant here unless dealing with natural numbers. Therefore, considering cases P and Q separately is the correct first step.

3) A case analysis proof must cover:

Most cases

All possible cases

Two cases only

Zero cases

In case analysis, it is crucial that all possible cases are covered to ensure that the conclusion holds universally. If even one case is missing, there might be a counterexample in that case, which would invalidate the proof. Covering most cases is insufficient because the missing case could be where the statement fails. There is no requirement that only two cases are used; the number of cases depends on the problem. Zero cases would mean no proof at all. Therefore, all possible cases must be covered.

Explanation

In case analysis, it is crucial that all possible cases are covered to ensure that the conclusion holds universally. If even one case is missing, there might be a counterexample in that case, which would invalidate the proof. Covering most cases is insufficient because the missing case could be where the statement fails. There is no requirement that only two cases are used; the number of cases depends on the problem. Zero cases would mean no proof at all. Therefore, all possible cases must be covered.

4) Which is a common case split for integers?

Positive, negative, zero

Only even

Only odd

Prime only

A common and exhaustive case split for integers is into positive, negative, and zero. This covers all integers because every integer is either positive, negative, or zero. Splitting only into even or odd is not exhaustive because it doesn't account for sign, but even/odd is common for parity proofs. However, for general integer properties, positive/negative/zero is a standard split. "Only even" or "only odd" are not exhaustive, and "prime only" misses many integers like composite numbers, negative integers, and zero. Thus, positive, negative, zero is a common and exhaustive split.

Explanation

A common and exhaustive case split for integers is into positive, negative, and zero. This covers all integers because every integer is either positive, negative, or zero. Splitting only into even or odd is not exhaustive because it doesn't account for sign, but even/odd is common for parity proofs. However, for general integer properties, positive/negative/zero is a standard split. "Only even" or "only odd" are not exhaustive, and "prime only" misses many integers like composite numbers, negative integers, and zero. Thus, positive, negative, zero is a common and exhaustive split.

5) Case analysis is also called:

Proof by contradiction

Proof by cases

Direct proof

Induction

Case analysis is synonymous with proof by cases. This technique involves dividing the proof into several cases and proving the statement in each case. Proof by contradiction is a different method where we assume the negation and derive a contradiction. Direct proof proves the statement without cases or contradiction. Induction is used for statements about natural numbers. Therefore, proof by cases is the other name for case analysis.

Explanation

Case analysis is synonymous with proof by cases. This technique involves dividing the proof into several cases and proving the statement in each case. Proof by contradiction is a different method where we assume the negation and derive a contradiction. Direct proof proves the statement without cases or contradiction. Induction is used for statements about natural numbers. Therefore, proof by cases is the other name for case analysis.

6) If a case analysis misses one case, the proof is:

Valid

Invalid

Stronger

Shorter

If a case analysis misses even one case, the proof is invalid because there might be a scenario in the missing case where the conclusion does not hold. A valid proof must establish the truth in all possible cases. If a case is missing, the proof is incomplete and cannot be trusted. It is not stronger or shorter; it is flawed. Therefore, the proof is invalid.

Explanation

If a case analysis misses even one case, the proof is invalid because there might be a scenario in the missing case where the conclusion does not hold. A valid proof must establish the truth in all possible cases. If a case is missing, the proof is incomplete and cannot be trusted. It is not stronger or shorter; it is flawed. Therefore, the proof is invalid.

7) Which statement best describes case analysis?

One continuous argument

Multiple sub-arguments

No arguments

Circular reasoning

Case analysis involves breaking down the proof into multiple sub-arguments, each corresponding to a different case. Each sub-argument proves the conclusion for that specific case. It is not one continuous argument because the cases are separate. It certainly involves arguments, so "no arguments" is incorrect. Circular reasoning is a fallacy and not descriptive of case analysis. Thus, multiple sub-arguments best describes case analysis.

Explanation

Case analysis involves breaking down the proof into multiple sub-arguments, each corresponding to a different case. Each sub-argument proves the conclusion for that specific case. It is not one continuous argument because the cases are separate. It certainly involves arguments, so "no arguments" is incorrect. Circular reasoning is a fallacy and not descriptive of case analysis. Thus, multiple sub-arguments best describes case analysis.

8) To prove “All real numbers x satisfy P(x),” a good case split is:

No cases

Cases x≥0, x<0

Only x>0

Only x<0

For real numbers, a common and exhaustive case split is into x≥0 and x<0. This covers all real numbers because every real number is either non-negative or negative. Using no cases would not constitute a case analysis. Only x>0 or only x<0 are not exhaustive because they miss x=0 or x≥0, respectively. Therefore, cases x≥0 and x<0 form a complete split.

Explanation

For real numbers, a common and exhaustive case split is into x≥0 and x<0. This covers all real numbers because every real number is either non-negative or negative. Using no cases would not constitute a case analysis. Only x>0 or only x<0 are not exhaustive because they miss x=0 or x≥0, respectively. Therefore, cases x≥0 and x<0 form a complete split.

9) What is the goal within each case?

Prove the hypothesis

Prove the conclusion holds

Find a counterexample

Assume the conclusion

Within each case of a case analysis, the goal is to prove that the conclusion holds under the assumptions of that case. The hypothesis is already assumed for the case, so proving it is unnecessary. Finding a counterexample would contradict the proof. Assuming the conclusion is circular reasoning. Therefore, we must prove that the conclusion follows from the case assumptions.

Explanation

Within each case of a case analysis, the goal is to prove that the conclusion holds under the assumptions of that case. The hypothesis is already assumed for the case, so proving it is unnecessary. Finding a counterexample would contradict the proof. Assuming the conclusion is circular reasoning. Therefore, we must prove that the conclusion follows from the case assumptions.

10) When proving “n² is even or odd,” you should split on:

N is even or odd

N is prime

N is positive

No split needed

To prove that n² is even or odd for an integer n, the most appropriate split is on whether n is even or odd. This is because the parity of n² depends on the parity of n: if n is even, n² is even; if n is odd, n² is odd. This split is exhaustive since every integer is either even or odd. Splitting on n being prime is not exhaustive because most integers are not prime. Splitting on n being positive misses negative integers, but parity applies to all integers. No split might work if directly proven, but case analysis using even/odd is straightforward and effective.

Explanation

To prove that n² is even or odd for an integer n, the most appropriate split is on whether n is even or odd. This is because the parity of n² depends on the parity of n: if n is even, n² is even; if n is odd, n² is odd. This split is exhaustive since every integer is either even or odd. Splitting on n being prime is not exhaustive because most integers are not prime. Splitting on n being positive misses negative integers, but parity applies to all integers. No split might work if directly proven, but case analysis using even/odd is straightforward and effective.

11) The statement 'For all real numbers x, x³ ≥ x' is:

Always true

True for x ≥ 1 and -1 ≤ x ≤ 0, false otherwise

True for x ≥ 0, false for x < 0

Never true

The statement is not universally true. For 0 < x < 1 (e.g., x = 0.5), x³ = 0.125 < 0.5 = x, making the statement false. For x ≥ 1 and -1 ≤ x ≤ 0, the inequality holds.

Explanation

The statement is not universally true. For 0 < x < 1 (e.g., x = 0.5), x³ = 0.125 < 0.5 = x, making the statement false. For x ≥ 1 and -1 ≤ x ≤ 0, the inequality holds.

12) Which case split is exhaustive for integers modulo 3?

{0,1}

{0,1,2}

{1,2}

{0}

For integers modulo 3, the possible remainders are 0, 1, and 2. Therefore, a case split based on these three values is exhaustive because every integer has a remainder of 0, 1, or 2 when divided by 3. The set {0,1} misses remainder 2, {1,2} misses remainder 0, and {0} misses 1 and 2. Thus, {0,1,2} is the exhaustive split.

Explanation

For integers modulo 3, the possible remainders are 0, 1, and 2. Therefore, a case split based on these three values is exhaustive because every integer has a remainder of 0, 1, or 2 when divided by 3. The set {0,1} misses remainder 2, {1,2} misses remainder 0, and {0} misses 1 and 2. Thus, {0,1,2} is the exhaustive split.

13) If a case analysis has 5 cases, you must prove the conclusion:

In at least 3 cases

In all 5 cases

In exactly 1 case

In no cases

In case analysis, regardless of the number of cases, you must prove the conclusion in every case. If the conclusion fails in even one case, the overall proof fails. Therefore, for 5 cases, you must prove the conclusion in all 5 cases. Proving in at least 3 or exactly 1 is insufficient. Proving in no cases would be no proof at all.

Explanation

In case analysis, regardless of the number of cases, you must prove the conclusion in every case. If the conclusion fails in even one case, the overall proof fails. Therefore, for 5 cases, you must prove the conclusion in all 5 cases. Proving in at least 3 or exactly 1 is insufficient. Proving in no cases would be no proof at all.

14) Which problem suits case analysis?

If-then with one hypothesis

Simple tautologies

Statements with multiple conditions

All proofs

Case analysis is particularly useful for statements with multiple conditions because it allows each condition to be handled separately. For example, if a statement depends on different cases like even/odd, positive/negative, etc., case analysis breaks down the proof into manageable parts. In contrast, if-then statements with one hypothesis can often be proven directly without cases, and simple tautologies may not require case analysis. While case analysis can be used in various proofs, it is not always necessary or efficient for all proofs, so "all proofs" is too broad. Thus, statements with multiple conditions are the most suitable.

Explanation

Case analysis is particularly useful for statements with multiple conditions because it allows each condition to be handled separately. For example, if a statement depends on different cases like even/odd, positive/negative, etc., case analysis breaks down the proof into manageable parts. In contrast, if-then statements with one hypothesis can often be proven directly without cases, and simple tautologies may not require case analysis. While case analysis can be used in various proofs, it is not always necessary or efficient for all proofs, so "all proofs" is too broad. Thus, statements with multiple conditions are the most suitable.

15) Which statement is true about case analysis?

Cases can be skipped

Cases can be arbitrary

Cases must be exhaustive

Cases should overlap heavily

The fundamental requirement for case analysis is that the cases must be exhaustive, meaning they cover all possible scenarios. Cases cannot be skipped because that would leave gaps in the proof. Cases should not be arbitrary; they should be chosen logically to cover the possibilities. Overlapping cases are not required and can be messy, so heavy overlap is not desirable. Therefore, cases must be exhaustive.

Explanation

The fundamental requirement for case analysis is that the cases must be exhaustive, meaning they cover all possible scenarios. Cases cannot be skipped because that would leave gaps in the proof. Cases should not be arbitrary; they should be chosen logically to cover the possibilities. Overlapping cases are not required and can be messy, so heavy overlap is not desirable. Therefore, cases must be exhaustive.

Fundamentals of Proof by Cases