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Logic + Sets
QuantSkills+Reasoning MATH 1001
| Question | Answer |
|---|---|
| Natural numbers | ℕ |
| Whole numbers | 𝕎 |
| Integers | ℤ |
| Rational numbers | ℚ |
| Real numbers | ℝ |
| Inductive reasoning | coming to a conclusion by looking at patterns or trends |
| Deductive reasoning | coming to a conclusion by applying general rules or procedures |
| Can a truth statement in math be both true and false? | No, it can only be true or be false |
| Counterexample | Scenario/object that proves a true statement false |
| Truth table | Shows all possible truth values for a simple statement or complex statement |
| ~ | negation; "not" |
| ^ | conjunction; "and" |
| ⌄ | disjunction; "or" |
| → | conditional; "if/then" |
| ⟷ | biconditional; "if and only if" |
| Do parenthesis matter when constructing logical statements? | Yes, parenthesis matter in logical statements |
| Existential qualifiers | some, many, few, at least one, there exist |
| Universal qualifiers | all, every, no, none |
| How should you negate a statement containing a universal qualifier? | Change the universal to an existential, add the "not"; and vice versa |
| How do you negate a statement containing a universal qualifier of "no/none" | Change the statement to contain "all/every," then convert to existential + add "not" |
| In an "or" truth table, when is the entire statement true? | If at least p or q is true |
| In an "and" truth table, when is the entire statement true? | If both p and q are true |
| In an "if/then" truth table, when is the entire statement true? | If p and q are both true, or if p is false |
| In an "if and only if" truth table, when is the entire statement true? | If p and q match truth values (T/T or F/F) |
| ≡ | logical equivalence, two statements have the same final outcome in a truth table |
| tautology | A statement that is always true |
| Self-contradiction | A statement that is always false |
| How do you apply De Morgan's Laws to logical statements? | Change an "and" to an "or," or vice versa, and negate both p and q |
| Is p→q logically equivalent to ~p⌄q? | Yes, they are logically equivalent |
| Set | Any group/collection of objects or values |
| { } or ø | Empty set |
| What method is {a, b, c}? | Roster method |
| What method is {x |x ∈ ℕ, x<10}? | Set builder notation |
| ∈ | Element of |
| ⊂ | Proper subset of |
| ⊆ | Subset of |
| Proper subset | A subset that is not equal to the set |
| Cardinality | The number of elements in a set; n(A) |
| How many subsets are in any given set? | 2^n |
| Well-defined set | A set in which it is clear what the elements consist of |
| Equal sets | When multiple sets have the exact same elements |
| Equivalent sets | When multiple sets have the same cardinality/number of elements |
| A' | Complement of set A; all elements within the universal set (U) that are not contained within A |
| What is the complement of the empty set? | The universal set |
| What is the only set with an odd number of subsets? | The empty set, 2^0 = 1 |
| ∩ | Intersection; the elements common between set A and set B |
| ∪ | Union; the combined elements of set A and set B |
| How is the answer to an intersection or union written? | As a set |
| Disjoint sets | Sets in which the intersection is the empty set (no elements in common) |
| Which logical connective is associated with intersection? | "and" |
| Which logical connective is associated with union? | "or" |
| What is the Inclusion-Exclusion Principle? | The total elements of the union between A and B is the sum of the number of elements in each, minus the number of elements in the intersection of A and B |
| What is the Percent Inclusion-Exclusion Principle? | The total percentage of the union between A and B is the sum of the percentages of both, minus the percentage of the intersection between A and B |
| How do you apply De Morgan's Laws to sets? | (A∪B)' = A'∩B' (A∩B)' = A' ∪ B' |
| Do parenthesis matter in sets? | Yes, parenthesis matter with sets |
| Commutative properties of sets | A∪B = B∪A A∩B = B∩A |
| Associative properties of sets | (A∪B)∪C = A∪(B∪C) (A∩B)∩C = A∩(B∩C) |
| When do the associative properties of sets work? | If all sets are either joined by only a union or only an intersection, must match |
| Distributive properties of sets | A∩(B∪C) = (A∩B)∪(A∩C) A∪(B∩C) = (A∪B)∩(A∪C) |
| Is { } an element of a set? | No, the empty set cannot be an element |
Created by:
MercuryDust
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