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Proof, EXT2 P1 2024 HSC 2 MC

Consider the following statement written in the formal language of proof

\(\forall \theta \in\biggl(\dfrac{\pi}{2}, \pi\biggr) \exists\ \phi \in\biggl(\pi, \dfrac{3 \pi}{2}\biggr) ; \ \sin \theta=-\cos \phi\).

Which of the following best represents this statement?

  1. There exists a \(\theta\) in the second quadrant such that for all \(\phi\) in the third quadrant  \(\sin \theta=-\cos \phi\).
  2. There exists a \(\phi\) in the third quadrant such that for all \(\theta\) in the second quadrant  \(\sin \theta=-\cos \phi\).
  3. For all \(\theta\) in the second quadrant there exists a \(\phi\) in the third quadrant such that  \(\sin \theta=-\cos \phi\).
  4. For all \(\phi\) in the third quadrant there exists a \(\theta\) in the second quadrant such that  \(\sin \theta=-\cos \phi\).
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\(C\)

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\(\Rightarrow C\)

Proof, EXT2 P1 2023 HSC 4 MC

Consider the following statement about real numbers.

"Whichever positive number \(r\) you pick, it is possible to find a number \(x\) greater than 1 such that

\(\dfrac{\ln x}{x^3}<r\). "

When this statement is written in the formal language of proof, which of the following is obtained?

  1. \(\forall x>1 \quad \exists r>0 \quad \dfrac{\ln x}{x^3}<r\)
  2. \(\exists x>1 \quad \forall r>0 \quad \dfrac{\ln x}{x^3}<r\)
  3. \(\forall r>0 \quad \exists x>1 \quad \dfrac{\ln x}{x^3}<r\)
  4. \(\exists r>0 \quad \forall x>1 \quad \dfrac{\ln x}{x^3}<r\)
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\(C\)

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\(\text{Whichever positive number}\ r\ \text{you pick …}\ \forall r>0 \)

\(\text{It is possible to find a number}\ x\ \text{greater then 1 …}\ \exists x>1 \)

\(\text{Such that}\ \ \dfrac{\ln x}{x^3}<r\)

\(\Rightarrow C\)

Proof, EXT2 P1 2021 HSC 12b

Consider Statement A.

Statement A: ‘If `n^2` is even, then `n` is even.’

  1. What is the converse of Statement A?.   (1 mark)

  2. Show that the converse of Statement A is true.   (1 mark)

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  1. `text{See Worked Solution}`
  2. `text{See Worked Solution}`
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i.    `text{Converse}`

`text{If} \ n \ text{is even, then} \ n^2 \ text{is even.}`
 

ii.   `text{If} \ n \ text{is even:}`

`n` `= 2p, \ p  ∈  ZZ`
`n^2` `= (2 p)^2`
  `=4 p^2`
  `= 2 (2p^2)`
  `=2q, \ q  ∈  ZZ`

 

`:. \ text{If} \ n \ text{is even, then} \ n^2 \ text{is even.}`

Proof, EXT2 P1 2021 HSC 5 MC

Which of the following statements is FALSE?

  1. `∀ a, b ∈ RR,`                                             `a < b \ => \ a^3 < b^3`
  2. `∀ a, b ∈ RR,`                                             `a < b \ => e^{-a} > e^{-b}`
  3. `∀ a, b ∈ (0, + ∞),`                               `a < b \ => \ text{ln} \ a < text{ln} \ b`
  4. `∀ a, b ∈ RR, text{with} \ a,b ≠ 0,`                    `a < b \ => \ 1/a > 1/b`
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`D`

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`text{By contradiction:}`

♦ Mean mark 50%.

`text{Consider} \ D`

`text{Let} \ \ a = -1 \ \ text{and} \ \ b = 1,`

`a < b \ -> \ -1 < 1 \ \ text{(TRUE)}`

`1/a > 1/b \ -> \ -1 > 1 \ \ text{(FALSE)}`
 

`=>\ D \ text{is false}`

Proof, EXT2 P1 2020 HSC 8 MC

Consider the statement:

"If `n` is even, then if `n` is a multiple of 3, then `n` is a multiple of 6".

Which of the following is the negation of this statement?

  1. `n` is odd and `n` is not a multiple of 3 or 6.
  2. `n` is even and `n` is a multiple of 3 but not a multiple of 6.
  3. If `n` is even, then `n` is not a multiple of 3 and `n` is not a multiple of 6.
  4. If `n` is odd, then if `n` is not a multiple of 3 then `n` is not a multiple of 6.
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`B`

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`text{Proposition: If} \ \ X =>Y`

♦♦ Mean mark 33%.

`X \ text{is a compound statement}`

`text{“If} \ n \ text{is even and a multiple of 3.”}`

`Y \ text{states “} n \ text{is a multiple of 6.”}`
 
`text{Negation if} \ X \ text{but} \ ¬ \ Y.`
 
`=> \ B`

Proof, EXT2 P1 EQ-Bank 2 MC

Consider the following statement.

"If you have no treasure, I have no kingdom."

Which of the following is logically equivalent to this statement?

  1.  If I have no kingdom then you have no treasure.
  2.  If you have treasure then I have a kingdom.
  3.  If you have no kingdom then I have no treasure.
  4.  If I have a kingdom then you have treasure.
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`D`

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`text(The statement is conditional.)`

`text(If)\ X\ text(then)\ Y\ \ text(or)\  \ X=>Y`

`text(The contrapositive statement is logically equivalent.)`

`text{(i.e.)}\ \ ¬Y => ¬X`

`text(If I have a kingdom then you have treasure.)`

`=>  D`