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| integers {..,-1,0,1,....} |
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Natural numbers
INCLUDES 0
{0, 1, 2, 3,...} |
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positive integers
{1, 2, 3, ...} |
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| real numbers - both rational and irrational |
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| rational numbers (fraction that involves two integers - ex 4/3) |
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⌊ x ⌋
Rounds DOWN function
floor(-3.75) = -4
floor (3.75) = 3 |
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⌈ x ⌉
Rounds UP function
ceiling(-3.75) = -3
ceiling(3.75) = 4 |
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| Notation for set membership, and not being in set membership |
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Use
x ∈ Z for x is an element of Z
Cross out ∈ for not an element - similar to not equals to. |
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statement that is either true or false (but never both)
example: 2 < 15
Cannot contain variables (ex: x < 3) |
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| truth table example for p ∧ q |
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"not p"
Example: Urbana is not in Illinois |
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True if only ONE is true.
See table:
[image] |
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| p implies q and conversely |
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negate both p and q, swap their roles
p → q
¬q → ¬p |
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• If it’s below zero, my car won’t start.
Converse and contrapositive this |
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• converse: If my car won’t start, it’s below zero
• contrapositive: If my car will start, then it’s not below zero. |
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| ¬p ∨ q is equivalent to ____ |
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| Logical Equivalence Symbol |
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≡
Give the same truth table/output |
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¬(p ∧ q) ≡ ¬p ∨ ¬q
¬(p ∨ q) ≡ ¬p ∧ ¬q
¬(¬p) ≡ p
Distributing in ¬ |
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p ∨ (q ∧ r) ≡
p ∧ (q ∨ r) ≡ |
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1. (p ∨ q) ∧ (p ∨ r)
2. (p ∧ q) ∨ (p ∧ r) |
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| ¬() around whole statement |
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| Statement that can be true or false |
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Expresses a range of values in a domain that make something true
Example in english: “some”,
“a couple”, “a few”, “many”, “most.” |
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With shorthand and english definitions:
universal quantifier
existential quantifier
unique existence quantifier
How to combine quantifiers? |
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universal - "for all" - ∀
existential - "there exists" - ∃
unique existence - "there is a unique" - ∃!
Combine with a comma. |
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