Using Systematic Casework

Hello every one

In the following example, I have a really problem to understand the third and thr fourth cases : May someone add explanation please and thank you ?

"If exactly one of these statements is false, which statement is false?

A. Statement D is true. B. Statement A is false. C. Statement B is false. D. Statement C is true.

Since one of the statements is false, it suffices to consider 4 cases, each case assuming one of statements to be false and then checking if this assumption is consistent with other statements or not.

Case 1: Statement A is false. This implies Statement D is false. Since no more than one statements can be false, this assumption is wrong.

Case 2: Statement B is false. This leads to conclusion that A is true, D is true, and C is true. (Looks good)

Case 3: Statement C is false. This results in conclusion of truthfulness of B, and thus fallacy of A, which is not possible.

Case 4: Statement D is false. It signifies that C is false, which is invalid.

So the statement which is wrong is Statement B." □

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1. numbered
2. list

numbered
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paragraph 1

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paragraph 1

paragraph 2

[example link](https://brilliant.org)

example link

> This is a quote

This is a quote

    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"

# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"

Math

Appears as

Remember to wrap math in $ ... $ or \[ ... \] to ensure proper formatting.

2 \times 3

2 \times 3

2^{34}

2^{34}

a_{i-1}

a_{i-1}

\frac{2}{3}

\frac{2}{3}

\sqrt{2}

\sqrt{2}

\sum_{i=1}^3

\sum_{i=1}^3

\sin \theta

\sin \theta

\boxed{123}

\boxed{123}

Easy Math Editor

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Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Perhaps it is the wording which is harder to understand?

In Case 3, we assumed that Statement C is false. So instead of "Statement B is false" we have "Statement B is true". So we now look at Statement B, which says that "Statement A is false". But we now have a problem, since we've already assumed that Statement C was false, and now we know that Statement A must also be false. So Case 3 doesn't work.

In Case 4, we assumed that Statement D was false. So instead of "Statement C is true" we have "Statement C is false". But again, this is a contradiction, since we already assumed that Statement D was false, and now we know that Statement C is false. So Case 4 also doesn't work.

Hope this helps.

David Stiff - 6 months, 2 weeks ago

Thank you very much for this helpful explanation. It s very kind of you. You re a right when you said my misunderstanding is because of the wording. It is also because I don t have neihter experience nor knowledge to solve this kind of problem. I m a perfect beginner.

Ahmed Chikh - 6 months, 2 weeks ago

You're very welcome! Language should not be a barrier to learning. Glad it helped!

May I ask you some recommendations so that guide me to the right way adopting to solve puzzles ; like readings, articles, videos ... ?

Probably the only place I could recommend is Khan Academy. They are a non-profit online learning website which has educational resources for all skill levels. However, sometimes searching for a specific topic on the Internet can be helpful as well. Brilliant.com also has some great wikis, like this one about solving logic puzzles. Just use the search bar at the top to find others.

Thank you very much.

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$