Lesson test: Proof technique

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Proof technique

1.[3p]
Match each proof technique to the rule that licenses it.

Direct proof
Proof by cases
Proof by contradiction
Existence proof

existential introduction
negation introduction
disjunction elimination
conditional proof

Show the answer

Direct proof: conditional proof Proof by cases: disjunction elimination Proof by contradiction: negation introduction Existence proof: existential introduction

2.[3p]
To prove "if $n^{2}$ is even then $n$ is even" by contraposition you show

that if $n$ is odd then $n^{2}$ is odd

Correct

that if $n$ is even then $n^{2}$ is even

The answer is: that if $n$ is odd then $n^2$ is odd

that some even $n$ has an even square

The answer is: that if $n$ is odd then $n^2$ is odd

that $n^{2}$ and $n$ are both even

The answer is: that if $n$ is odd then $n^2$ is odd

3.[2p]
Proving the converse of a claim

proves a different statement and establishes nothing about the original

Correct

is the same as proving the contrapositive

The answer is: proves a different statement and establishes nothing about the original

proves the original claim

The answer is: proves a different statement and establishes nothing about the original

is only valid for biconditionals

The answer is: proves a different statement and establishes nothing about the original

4.[3p]
The claim that every odd composite number is a prime plus twice a square first fails at which number?

CorrectNot quite: 5777

5.[2p]
How many counterexamples are needed to refute a universal claim?

One

Correct

One for each case

The answer is: One

Enough to establish a pattern

The answer is: One

As many as there are objects in the domain

The answer is: One

6.[2p]
Induction is a rule of first-order logic.

True

The answer is: False

False

Correct

7.[3p]
Why is induction not derivable from the rules of this course?

A derivation is finite, so it reaches $S (1)$ , $S (2)$ and so on one at a time and never all of them

Correct

The inductive step is invalid

The answer is: A derivation is finite, so it reaches $S(1)$, $S(2)$ and so on one at a time and never all of them

The base case cannot be proved

The answer is: A derivation is finite, so it reaches $S(1)$, $S(2)$ and so on one at a time and never all of them

Universal introduction is restricted to finite domains

The answer is: A derivation is finite, so it reaches $S(1)$, $S(2)$ and so on one at a time and never all of them

8.[3p]
The proof that irrational $a$ and $b$ exist with $a^{b}$ rational, arguing from ${\sqrt{2}}^{\sqrt{2}}$ , is

non-constructive, because it does not say which of the two cases holds

Correct

invalid, because it never produces a number

The answer is: non-constructive, because it does not say which of the two cases holds

constructive, because it names $\sqrt{2}$

The answer is: non-constructive, because it does not say which of the two cases holds

circular

The answer is: non-constructive, because it does not say which of the two cases holds

9.[2p]
The cases in an argument by cases must be exhaustive but need not be mutually exclusive.

True

Correct

False

The answer is: True

1.[3p] Match each proof technique to the rule that licenses it.

2.[3p] To prove "if n2 is even then n is even" by contraposition you show

3.[2p] Proving the converse of a claim

4.[3p] The claim that every odd composite number is a prime plus twice a square first fails at which number?

5.[2p] How many counterexamples are needed to refute a universal claim?

6.[2p] Induction is a rule of first-order logic.

7.[3p] Why is induction not derivable from the rules of this course?

8.[3p] The proof that irrational a and b exist with ab rational, arguing from 22, is

9.[2p] The cases in an argument by cases must be exhaustive but need not be mutually exclusive.

1.[3p]
Match each proof technique to the rule that licenses it.

2.[3p]
To prove "if $n^{2}$ is even then $n$ is even" by contraposition you show

3.[2p]
Proving the converse of a claim

4.[3p]
The claim that every odd composite number is a prime plus twice a square first fails at which number?

5.[2p]
How many counterexamples are needed to refute a universal claim?

6.[2p]
Induction is a rule of first-order logic.

7.[3p]
Why is induction not derivable from the rules of this course?

8.[3p]
The proof that irrational $a$ and $b$ exist with $a^{b}$ rational, arguing from ${\sqrt{2}}^{\sqrt{2}}$ , is

9.[2p]
The cases in an argument by cases must be exhaustive but need not be mutually exclusive.