Laws of Logic Flashcards
If A and B are disjoint, this means…
A∩B=∅
~(P ^ Q) is equalivent to…
~P v ~Q
~(P v Q) is equalivent to…
~P ^ ~Q
Are P v Q and P ^ Q commutative?
Yes.
Are P ^ (Q ^ R) and P v (Q v R) associative?
Yes.
P v P and P ^ P is equalivent to…
P
P ^ (Q v R) and P v (Q ^ R) are equalivent to…
(P ^ Q) v (P ^ R) and (P v Q) ^ (P v R) respectively.
P v ( P ^ Q) and P ^ (P v Q) are equal to…
P
~(~P) is equal to…
P
If Δ is defined as the symmetric difference operator, then A Δ B is equalivent to…
(A U B) \ (A n B)
or
(A \ B) U (B \ A)
A \ B is equalivent to…
A \ B = {x |x ∈ A ^ x !∈ B}
P → Q, is equalivent to…
~P v Q
or
~(P ^ ~Q)
P ↔ Q is equalivent to…
(P → Q) ∧ (¬P → ¬Q)
or
(~P v Q) ^ (P v ~Q)
A ⊆ B is equalivent to…
∀x(x ∈ A → x ∈ B)
∀x P(x) means…
For all values of x, P(x) is true, or in other words:
P(x) is universally true.
P(x) = U, where U is the universal set.
What does ∀x∈A,P(x) mean?
∀x(x∈A→P(x))
∃x P(x) means…
There exists one value of x in the solution set of P(x), or that:
the solution set of P(x) does not equal ∅.
What does ∃!xP(x) mean?
There exists exactly one x such that P(x) is true.
What does ∃x∈A,P(x) mean?
∃x(x∈A∧P(x))
¬∃xP(x) and ¬∀xP(x) are equalivent to…
¬∃xP(x)is equivalent to∀x¬P(x).
¬∀xP(x)is equivalent to∃x¬P(x).
Suppose A = {1,2,3}, B = {4}, and C = ∅
If set F = {A, B,C}, does 1 ∈ F?
No.
1 is not in the set of F, but in the set of A.
What is a power set? ℙ(A)
Suppose A is a set.
The power set of A, denoted ℙ(A), is
the set whose elements are all the subsets of A. In other words,
ℙ(A) = {x |x ⊆ A}.
For example, the set A = {7,12} has four subsets: {∅}, {7}, {12}, and {7,12}.
Thus, ℙ(A) = { ∅ , {7},{12},{7,12}}
∩F = ?
∪F = ?
∩F = {x | ∀A ∈ F(x ∈ A)} = {x | ∀A(A ∈ F → x ∈ A)}
∪F = {x | ∃A ∈ F(x ∈ A)} = {x | ∃A(A ∈ F ∧ x ∈ A)}
