Logic

Negation and Logical Equivalence

100

Write the negation of: "The server is running or the database is corrupted."      

"The server is not running and the database is not corrupted." (Or: ∼p ∧ ∼q)      

200

Are ∼(p ∨ q) and ∼p ∧ ∼q logically equivalent? Yes or no, and name the law.      

Yes. De Morgan's Law.      

300

Write the negation of: "If the alarm sounds, then the building is evacuated."      

"The alarm sounds and the building is not evacuated." (Or: p ∧ ∼q)      

400

Negate: "All students submitted their assignments on time."      

"There exists at least one student who did not submit their assignment on time." (Or: ∃x, ∼S(x))      

500

Write the negation of: "3 < y ≤ 8"      

"y ≤ 3 or y > 8"      

Conditionals and Contrapositives

100

Write the contrapositive of: "If it is raining, then the ground is wet."

"If the ground is not wet, then it is not raining."

200

Is the contrapositive of a conditional statement logically equivalent to the original? Yes or no

Yes

300

Write the converse of: "For all real numbers x, if x > 3 then x² > 9."

"For all real numbers x, if x² > 9 then x > 3."

400

Affirming the consequent

Affirming the consequent

500

Write the contrapositive of: "If Ann is Jan's mother, then Jose is Jan's cousin."

"If Jose is not Jan's cousin, then Ann is not Jan's mother."

Quantifiers and predicates

100

What quantifier symbol means "there exists"?

∃ (existential quantifier)

200

Formally write using quantifiers: "Every rational number can be written as a ratio of two integers.

∀x ∈ ℚ, ∃a, b ∈ ℤ such that x = a/b (or similar formal notation)

300

Write the negation using quantifiers: "All dogs are loyal."

∃x, D(x) ∧ ∼L(x) where D(x) = "x is a dog" and L(x) = "x is loyal"

400

Use predicates and quantifiers to negate: "Having a large income is a necessary condition for happiness."

∃x, I(x) ∧ ∼H(x) where I(x) = "x has a large income" and H(x) = "x is happy"

500

Formally write: "For all real numbers x, if x < 1 then 1/x > 1" and then negate it.

Original: ∀x ∈ ℝ, x < 1 → 1/x > 1 Negation: ∃x ∈ ℝ, x < 1 ∧ 1/x ≤ 1

Argument Validity

100

Name the valid argument form: "If p then q; p is true; therefore q is true."

Modus ponens.

200

What is the name of the valid argument form: "If p then q; q is false; therefore p is false"?

Modus tollens

300

Is this argument valid? "All cheaters sit in the back row. Monty sits in the back row. Therefore, Monty is a cheater." Yes or no, and why.

No. It commits the fallacy of affirming the consequent. Just because Monty sits in the back row doesn't mean he's a cheater; there could be other reasons to sit there

400

Determine validity: "If the network is down, then users cannot access the database. Users cannot access the database. Therefore, the network is down."

Invalid. This is the fallacy of affirming the consequent. Users not accessing the database could have other causes besides the network being down.

500

What logical rule allows us to apply a universal statement to a specific individual?

Universal instantiation.

Number Base conversions

100

Convert 101011₂ to decimal.

43₁₀

200

Convert 63₁₀ to binary.

111111₂

300

Convert 3B7₁₆ to decimal.

951₁₀

400

Convert 88₁₀ to binary.

1011000₂

500

Convert 512₁₀ to hexadecimal

200₁₆