QUESTIONS ANSWERS

(1)

European section, season 2 Monday, october the 16th, 2009

Test #1

Some items of this test are multiple choice questions. A good answer earns 2 points and any wrong answer costs 1 point, a missing answer earns or costs nothing. Other items are free response questions, all of them worth 4 points, where any incomplete or imperfect answer will be rewarded.

Section 1 – Syllogisms

QUESTIONS ANSWERS

1.Isthefollowingsyllogismvalid, true,neitherofboth?

Nohorsesareblue.

Somebirdsareblue.

Nobirdsarehorses.

❒

^True

❒

^Valid

❒

^Neither

❒

^Both

2.Isthefollowingsyllogismvalid, true,neitherofboth?

Drivingabigarusesalotofgas.

Usingalotofgasisexpensive.

Drivingabigarisexpensive.

❒

^True

❒

Valid

❒

^Neither

❒

^Both

3.Whatisthemiddletermin thefollowingsyllogism?

All sunnydaysaregreat.

Somemondaysaresunnydays.

SomeMondaysaregreat.

❒

^Sunny^days

❒

^Great

❒

^Mondays

❒

^Are

4.Whatisthetotalnumberofvalid typesofsyllogisms.

❒

²⁵⁶

❒

¹⁹

❒

⁴

❒

⁴²

5.TheletterA standsfor

❒

^Universal^armative.

❒

^Universal^negative.

❒

^Partiular^armative.

❒

^Partiular^negative.

6

^. ^Writeâ^Barbara^syllogismôf^yourôwnînvention.

(2)

Section 2 – Formal logic and equivalences 1

^. ^Math^eah^logial^operator^to ^the^right^truth^table.

p ∧ q p ∨ q p → q p ↔ q

❒ ❒ ❒ ❒

❒ ❒ ❒ ❒ ❒ ❒

p q 0 0 1 0 1 0 1 0 0 1 1 1

p q 0 0 1 0 1 0 1 0 1 1 1 1

p q 0 0 0 0 1 1 1 0 1 1 1 1

p q 0 0 0 0 1 0 1 0 0 1 1 1

p q 0 0 1 0 1 1 1 0 0 1 1 1

p q 0 0 0 0 1 0 1 0 1 1 1 1

QUESTIONS ANSWERS

2.Theproposition

p → q

^is^formally^equivalent^to

❒ q → p

❒ ¬q → ¬p

❒ ¬p → ¬q

❒ p ↔ q

3.Theproposition

¬(p ∨ q)

❒ p ∧ q

❒ ¬p ∧ ¬q

❒ ¬p ∨ ¬q

❒ p ∨ q

4.Theproposition

¬(p ∧ q)

❒ p ∧ q

❒ ¬p ∧ ¬q

❒ ¬p ∨ ¬q

❒ p ∨ q

5.Theproposition

p ↔ q

❒ (p → q) ∧ (q → p)

❒ (p → q) ∨ (q → p)

❒ (p → q) ∧ (p → ¬q)

❒ (p → q) ∨ (p → ¬q) 6

^. ^Prove,^by^building^their^truth^tables,^that^the^followingpropositionsareformallyequivalent.

¬(r → p ∨ q)

^and

¬p ∧ ¬q ∧ r.

(3)

Section 3 – Venn diagrams 1

^. ^Draw ^a^Venn ^diagram ^with ^two^sets

A

^and

B

^suh^that

x ∈ A → x ∈ B

^.

2

^. ^Draw^a ^V^enn ^diagram ^with ^three ^sets

A

^,

B

and

C

^suh^that

A ⊂ B

^and

C ⊂ B

^.

The following multiple choice questions are all about the configuration you represented in the question 2.

QUESTIONS ANSWERS

3.Theintersetion

A ∩ B

^is ^equal^to

❒ ∅

❒ A

❒ B

❒

^Neither^of ^these^sets.

4.Theintersetion

A ∩ C

^is^equal^to

❒ ∅

❒ A

❒ B

❒

5.Theintersetion

B ∩ C

^is^equal^to

❒ ∅

❒ A

❒ B

❒

6.Istheequality

A ∪ B = A

^true^of^falseⁱⁿ^this^situation^?

❒

^True

❒

False

❒

It'simpossibletosay

7.Istheequality

A ∪ B = B

^true^of^falseⁱⁿ^this^situation^?

❒

^True

❒

^F^alse

❒

^It's^impossible^to^say

8.Istheinlusion

A ∪ C ⊂ B

^true^of^falseⁱⁿ^this ^situation^?

❒

^True

❒

^F^alse

❒

9.Istheequality

B ∪ C = Ω

^true^of^falseⁱⁿ^this ^situation^?

❒

^True

❒

^F^alse

❒

10.Is theequality

B ∪ A = Ω

^true^of^falseⁱⁿ ^this^situation^?

❒

^True

❒

^F^alse

❒

(4)

Section 4 – Set-builder notation 1

^. ^Mathêah^set^denition ^with^the^right^partial ^listôfêlements.

{n ² : n ∈

^N

∩ [2, 7]} {x ∈

^R

: x ² = 2x} {3k + 1 : k ∈

^Z

} {n : n ∈

^Z

∧ 15 n ∈

^Z

}

❒ ❒ ❒ ❒

2

^,

0 16

^,

37

^,

10 16

^,

36

^,

9 1

^,

3

^,

15 QUESTIONS ANSWERS

2.Whihoneofthefollowingsetsis empty?

❒ {n ∈

^N

: n ² = 4}

❒ {n ∈

^N

: n ² = 5}

❒ {n ∈

^R

: n ² = 4}

❒ {n ∈

^R

: n ² = 5}

3.ThenotationR

+

denoteswhatset?

❒

N

❒ ]0; +∞[

❒ [0; +∞[

❒

^Q

⁺

4.Thenaturalnumbersaretheelementsofwhatset?

❒

^N

❒

N

⋆

❒

^Z

❒

^Z

^⋆

5.Onlyoneofthefollowingstatementsistrue.Whihone?

❒ ∀x ∈

^R,

x ² = 5

❒ ∀x ∈

^Z,

x ² = 5

❒ ∃x ∈

^R,

x ² = 5

❒ ∃x ∈

^Z,

x ² = 5

6

^. ^Explainⁱⁿ ^a^few^words^Russell's^paradox.