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Math
Discrete Math
Discrete Mathematics Exam 1
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Terms in this set (40)
p v q
This is a disjunction: p or q, or p and q. This is an inclusive or.
p ^ q
This is a conjunction: p and q
p ⊕ q
This is an exclusive or: either p or q
p → q
This is an implication. If p, then q
Converse conditional statements
q → p
Contrapositive conditional statements
¬q → ¬p (has the same truth values as p → q)
Inverse conditional statement
¬p → ¬q
p ↔ q
This is a biconditional statement, also known as bi-implications. It means p if and only if q. True if both p and q have the same truth values. Also written as "p is necessary and sufficient for q", "if p then q, and conversely", and "p iff q".
Precedence of logical operators in 1st to 5th
1. ¬ 2. ^ 3. v 4. → 5. ↔
Bit
This is a symbol with two possible values, specifically 0 (zero) and 1 (one). 1 represents the True value and 0 represents a False value.
De Morgan's law
When you distribute a "¬", then you flip the conjunction or disjunction sign that you are distributing to.
p ∧ T ≡ p
p ∨ F ≡ p
Identity laws
p ∨ T ≡ T
p ∧ F ≡ F
Domination laws
p ∨ p ≡ p
p ∧ p ≡ p
Idempotent laws
¬(¬p) ≡ p
Double negation law
p ∨ q ≡ q ∨ p
p ∧ q ≡ q ∧ p
Commutative laws
(p ∨ q) ∨ r ≡ p ∨ (q ∨ r)
(p ∧ q) ∧ r ≡ p ∧ (q ∧ r)
Associative laws
p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
Distributive laws
¬(p ∧ q) ≡ ¬p ∨ ¬q
¬(p ∨ q) ≡ ¬p ∧ ¬q
De Morgan's laws (1st and 2nd)
p ∨ (p ∧ q) ≡ p
p ∧ (p ∨ q) ≡ p
Absorption laws
p ∨ ¬p ≡ T
p ∧ ¬p ≡ F
Negation laws
Predicate
refers to a property that the subject of the statement can have (e.g. is greater than 3)
Propositional function
P at x or P(x); the function
Functionally Complete
Every compound proposition is logically equivalent to a compound proposition composed of only these logical operators
Satisfiability
Occurs when there is an assignment of truth values to its variables that makes it true
Set N
natural numbers = {0,1,2,3...}
Set Z
integers = {....-3,-2,-1,0,1,2,3....}
Set Z+
positive integers = {1,2,3.....}
Set R
real numbers
set R+
positive real numbers
set C
complex numbers
Q
set of rational numbers
Injection
Every single input into A (the domain) has a single output in B (the codomain). However, all of the codomain values do not have to be matched up. "One-to-one"
Surjection
Every element of the codomain is matched up with a value from the domain.
Bijection
It is both injective and surjective. In other words, every one of the domain has a single single output and every one of the codomain values is matched up with a value of the domain.
א
The Hebrew symbol aleph. |S| is equal to the cardinality "aleph null"
Join of matrices
A v B
The meet of matrices
A ^ B
Boolean Product
^ between terms and then v between the two deciding terms
Boolean Powers of zero-one matrices
for all positive integers n with n >=5, the matrix becomes all ones
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