First Week Questions
What does it mean if b|a?
We say that a divides b if there is an integer c such that b = a · c
How would you show two logic statements are equivalent?
Truth Table!
What is the definition of a set?
A set is an unordered collection of distinct objects.
What does ∃ mean generally?
There is/There exists (Singular)
When does |A ∪ B| = |A| + |B|?
When A and B are disjoint
What does it mean if an integer is odd?
An integer a is called odd if there is an integer x such that a = 2x + 1
What is the truth table for P implies Q?
T
F
T
T
Write the set of positive even integers in set builder notation
S = { xeZ: 2|x and x>0}
What does ∀ mean generally?
For all (every possible option)
What is the Well Ordering Principle?
Every nonempty set S of natural number has a smallest element.
How do you prove A iff B?
(⇒) prove (directly) that “If A, then B”. ‹ (⇐) prove (directly) that “If B, then A”.
What is one of the two the Distributive Laws?
P ∧ (Q ∨ R) ⇔ (P ∧ Q) ∨ (P ∧ R) or P ∨ (Q ∧ R) ⇔ (P ∨ Q) ∧ (P ∨ R)
What does A ∪ (B ∩ C) simplify too?
(A∪B)∩ (A∪C)
Translate this sentence with quantifiers, ”No matter what integer you choose, there is always another integer that is larger”
∀x ∈ Z, ∃y∈ Z: y>x
What is the Euclidean division algorithm?
Let a and b be integers with b > 0. Then there are unique integers q and r such that a = bq + r and 0 ≤ r < b.
How do you disprove a statement?
Provide a counterexample!
What is one of the two D'Morgan's Laws?
¬(P ∨ Q) ⇔ (¬P) ∧ (¬Q) or ¬(P ∧ Q) ⇔ (¬P) ∨ (¬Q)
What is the Symmetric Difference between A and B?
(A - B) ∪ (B - A) or
(A ∪ B) - (A ∩ B)
What is the negation of ∀x ∈ Z : x < 0?
∃x ∈ Z: x >= 0
What is the complement of a set A?
U - A