Direct Proof Quiz: Strengthen Step-by-Step Logical Arguments

1) In proving "If a relation R is symmetric and transitive, and every element is related to at least one element, then R is reflexive" directly, we need to:

Assume aRa is false for some a

Prove the contrapositive statement

Use induction on the number of elements

For each a, find b such that aRb, then use symmetry and transitivity to show aRa

From aRb and symmetry, bRa; from bRa and aRb, transitivity yields aRa.

If every element a is related to some b and R is symmetric, then aRb implies bRa. Using transitivity on aRb and bRa shows aRa, establishing reflexivity.

Explanation

From aRb and symmetry, bRa; from bRa and aRb, transitivity yields aRa.

If every element a is related to some b and R is symmetric, then aRb implies bRa. Using transitivity on aRb and bRa shows aRa, establishing reflexivity.

2) In a direct proof that "If a|b and a|c, then a|(b+c)," which is the appropriate formalization of the premises?

Let b+c = an for some integer n

Assume a does not divide b or c

Suppose a does not divide b+c

Let b = ak and c = am for integers k,m

Express b and c as multiples of a: b = ak and c = am, then b + c = a(k + m).

Since a divides b and c, we can write them as ak and am. Adding gives b + c = a(k + m), which is a multiple of a, so a divides b + c.

Explanation

Express b and c as multiples of a: b = ak and c = am, then b + c = a(k + m).

Since a divides b and c, we can write them as ak and am. Adding gives b + c = a(k + m), which is a multiple of a, so a divides b + c.

3) Which approach is appropriate for a direct proof of "If n is divisible by 6, then n is divisible by 2 and 3"?

Assume n is not divisible by 2 or 3

Prove the contrapositive statement

Use mathematical induction on k

Let n = 6k, then show n = 2(3k) and n = 3(2k)

From n = 6k, factor 6 as 2·3 to show n is divisible by both 2 and 3.

If n = 6k, then n = 2(3k) and n = 3(2k), so n is simultaneously divisible by 2 and by 3.

Explanation

From n = 6k, factor 6 as 2·3 to show n is divisible by both 2 and 3.

If n = 6k, then n = 2(3k) and n = 3(2k), so n is simultaneously divisible by 2 and by 3.

4) True or False: In a direct proof, we assume the conclusion is true and work backwards.

True

False

In a direct proof, we assume the hypothesis (P) and derive the conclusion (Q).

A direct proof starts from the given condition P and uses logical steps to reach Q without using contradiction or contrapositive.

Explanation

In a direct proof, we assume the hypothesis (P) and derive the conclusion (Q).

A direct proof starts from the given condition P and uses logical steps to reach Q without using contradiction or contrapositive.

5) In a direct proof of "If it is raining, then the ground is wet," we start by assuming:

It is not raining

The ground is wet

The ground is not wet

It is raining

To prove an “if–then” statement, we begin by assuming the given condition is true and then show that the result logically follows.

Explanation

To prove an “if–then” statement, we begin by assuming the given condition is true and then show that the result logically follows.

6) Which is a valid direct proof approach for "If x² = y² and x,y > 0, then x = y"?

Assume x ≠ y and derive contradiction

Prove the contrapositive statement

Use mathematical induction

Since x² - y² = 0, we have (x-y)(x+y) = 0, so x-y = 0 (since x+y > 0)

Factor x² − y² and use x + y > 0 to conclude x − y = 0, hence x = y.

Since x² = y², factoring gives (x − y)(x + y) = 0. Because x and y are positive, x + y > 0, so the only zero factor is x − y, giving x = y.

Explanation

Factor x² − y² and use x + y > 0 to conclude x − y = 0, hence x = y.

Since x² = y², factoring gives (x − y)(x + y) = 0. Because x and y are positive, x + y > 0, so the only zero factor is x − y, giving x = y.

7) In proving "If a and b are both odd, then a+b is even" directly, we should:

Assume a+b is odd

Prove the contrapositive statement

Use mathematical induction on a+b

Let a = 2k+1 and b = 2m+1, then a+b = 2(k+m+1)

Express odd numbers as 2k+1 and simplify their sum to an even form.

Writing a = 2k+1 and b = 2m+1 shows a + b = 2(k + m + 1), which is even because it is 2 times an integer.

Explanation

Express odd numbers as 2k+1 and simplify their sum to an even form.

Writing a = 2k+1 and b = 2m+1 shows a + b = 2(k + m + 1), which is even because it is 2 times an integer.

8) Which is the correct first step in a direct proof that "If A and B are countable sets, then A∪B is countable"?

Assume A∪B is uncountable

Prove the contrapositive statement

Use induction on the size of A and B

List the elements of A as a₁,a₂,... and B as b₁,b₂,...

Start from enumerations of A and B to build an enumeration of A ∪ B.

If A and B are countable, they can be listed as sequences. By interweaving these sequences, we obtain a sequence listing all elements of A ∪ B.

Explanation

Start from enumerations of A and B to build an enumeration of A ∪ B.

If A and B are countable, they can be listed as sequences. By interweaving these sequences, we obtain a sequence listing all elements of A ∪ B.

9) True or False: A direct proof of "If P then Q" involves assuming P is true.

True

False

Direct proofs assume the hypothesis and derive the conclusion.

The method relies on forward reasoning: begin with what is given and step logically to what must follow.

Explanation

Direct proofs assume the hypothesis and derive the conclusion.

The method relies on forward reasoning: begin with what is given and step logically to what must follow.

10) In a direct proof, after assuming the hypothesis, our goal is to:

Find a contradiction

Prove the negation of the hypothesis

Relevant with examples

Derive the conclusion through logical steps

We use logical steps from the hypothesis to reach the conclusion.

Once we assume the hypothesis, each step must be justified so that the final statement proves the desired conclusion.

Explanation

We use logical steps from the hypothesis to reach the conclusion.

Once we assume the hypothesis, each step must be justified so that the final statement proves the desired conclusion.

11) In a direct proof that "If f is increasing and g is increasing, then f∘g is increasing," we begin by:

Assuming f(g(x)) < f(g(y))

Letting h = f∘g and assuming h is increasing

Assuming x < y and showing f(g(x)) < f(g(y))

Supposing f∘g is not increasing

Start with x
If x

Explanation

Start with x
If x

12) For proving "If a ring R has no zero divisors, and ab = ac with a ≠ 0, then b = c" directly, the key step is:

Assume b ≠ c and find contradiction

Prove the contrapositive statement

Use induction on the number of ring elements

Rewrite as a(b-c) = 0, then use no zero divisors property

Factor the equation and apply the no-zero-divisors property to deduce b = c.

From ab = ac, rewrite as a(b − c) = 0. Since a ≠ 0 and the ring has no zero divisors, b − c must equal 0, giving b = c.

Explanation

Factor the equation and apply the no-zero-divisors property to deduce b = c.

From ab = ac, rewrite as a(b − c) = 0. Since a ≠ 0 and the ring has no zero divisors, b − c must equal 0, giving b = c.

13) Which is a valid direct proof approach for "If a|b and b|c, then a|c"?

Assume a does not divide c

Prove the contrapositive statement

Use mathematical induction on c

Since b = ak and c = bm, we have c = a(km), so a|c

Express b and c as multiples of a and b, then combine the expressions.

If b = ak and c = bm, then substituting b gives c = a(km). This shows c is a multiple of a, so a divides c.

Explanation

Express b and c as multiples of a and b, then combine the expressions.

If b = ak and c = bm, then substituting b gives c = a(km). This shows c is a multiple of a, so a divides c.

14) Direct proof is the only valid proof technique in mathematics.

True

False

Other techniques include contradiction, contrapositive, induction, etc.

Direct proof is not the only method; many results are easier or only possible using other proof strategies.

Explanation

Other techniques include contradiction, contrapositive, induction, etc.

Direct proof is not the only method; many results are easier or only possible using other proof strategies.

15) In proving "If f and g are continuous at x = c, then f+g is continuous at x = c" directly, we start by:

Assuming f+g is continuous at c

Proving the contrapositive statement

Using proof by contradiction

Using the definition of continuity for f and g at c

Use ε–δ definitions for f and g to show continuity of f + g.

Given ε > 0, use continuity of f and g to find δ-values that control their changes. Combine these to show |(f+g)(x) − (f+g)(c)| becomes arbitrarily small.

Explanation

Use ε–δ definitions for f and g to show continuity of f + g.

Given ε > 0, use continuity of f and g to find δ-values that control their changes. Combine these to show |(f+g)(x) − (f+g)(c)| becomes arbitrarily small.

16) In a direct proof that "If a|b and b|c, then a|c" (where | means divides), which is the first appropriate statement?

Assume a|c

Assume a|b and b|c

Let b = ak and c = bm for integers k,m

Let c = an for some integer n

Begin by assuming the hypothesis that a divides b and b divides c.

Start from b = ak and c = bm for integers k and m, then work toward proving a divides c.

Explanation

Begin by assuming the hypothesis that a divides b and b divides c.

Start from b = ak and c = bm for integers k and m, then work toward proving a divides c.

17) To directly prove "If x > y > 0, then x² > y²," the first step is to:

Assume x² > y²

Write x = y + d where d > 0

Assume x > y > 0

Factor x² - y² = (x-y)(x+y)

Start from the hypothesis x > y > 0 before manipulating algebraically.

Once we assume x > y, we can multiply by the positive quantity (x + y) to show x² > y².

Explanation

Start from the hypothesis x > y > 0 before manipulating algebraically.

Once we assume x > y, we can multiply by the positive quantity (x + y) to show x² > y².

18) In the direct proof that "If m and n are odd integers, then m+n is even," which statement correctly formalizes the premise?

Let m+n = 2p for some integer p

Assume m and n are not even

Suppose m+n is not odd

Let m = 2k+1 and n = 2l+1 for integers k,l

Expressing odd integers as 2k+1 and 2l+1 is the standard formalization.

Using this form makes it easy to show their sum, product, or other expressions maintain expected parity properties.

Explanation

Expressing odd integers as 2k+1 and 2l+1 is the standard formalization.

Using this form makes it easy to show their sum, product, or other expressions maintain expected parity properties.

19) Which would be an appropriate first step in a direct proof of "If 3x+2 = 8, then x = 2"?

Assume 3x+2 = 8

Subtract 2 from both sides of 3x+2 = 8

Assume x = 2

Substitute x = 2 into 3x+2

Start from the given equation and then solve for x.

Assume 3x + 2 = 8 and isolate x through algebraic steps to obtain x = 2.

Explanation

Start from the given equation and then solve for x.

Assume 3x + 2 = 8 and isolate x through algebraic steps to obtain x = 2.

20) In proving "If n is odd, then n² is odd" directly, which is the correct formalization of the premise?

Assume n is not even

Suppose n² is not even

Let n² = 2m+1 for some integer m

Let n = 2k+1 for some integer k

Express an odd integer as 2k+1, then square it and simplify.

Squaring gives n² = (2k+1)² = 4k² + 4k + 1 = 2(2k² + 2k) + 1, which is again odd.

Explanation

Express an odd integer as 2k+1, then square it and simplify.

Squaring gives n² = (2k+1)² = 4k² + 4k + 1 = 2(2k² + 2k) + 1, which is again odd.