Negation and Logical Equivalence
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)
Write the contrapositive of: "If it is raining, then the ground is wet."
"If the ground is not wet, then it is not raining."
What quantifier symbol means "there exists"?
∃ (existential quantifier)
Name the valid argument form: "If p then q; p is true; therefore q is true."
Modus ponens.
Convert 101011₂ to decimal.
43₁₀
Are ∼(p ∨ q) and ∼p ∧ ∼q logically equivalent? Yes or no, and name the law.
Yes. De Morgan's Law.
Is the contrapositive of a conditional statement logically equivalent to the original? Yes or no
Yes
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)
What is the name of the valid argument form: "If p then q; q is false; therefore p is false"?
Modus tollens
Convert 63₁₀ to binary.
111111₂
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)
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."
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"
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
Convert 3B7₁₆ to decimal.
951₁₀
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))
Affirming the consequent
Affirming the consequent
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"
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.
Convert 88₁₀ to binary.
1011000₂
Write the negation of: "3 < y ≤ 8"
"y ≤ 3 or y > 8"
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."
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
What logical rule allows us to apply a universal statement to a specific individual?
Universal instantiation.
Convert 512₁₀ to hexadecimal
200₁₆