Laws of Logic

Question 1

If A and B are disjoint, this means…

Answer 1

A∩B=∅

Question 2

~(P ^ Q) is equalivent to…

Answer 2

~P v ~Q

Question 3

~(P v Q) is equalivent to…

Answer 3

~P ^ ~Q

Question 4

Are P v Q and P ^ Q commutative?

Answer 4

Yes.

Question 5

Are P ^ (Q ^ R) and P v (Q v R) associative?

Answer 5

Yes.

Question 6

P v P and P ^ P is equalivent to…

Answer 6

P

Question 7

P ^ (Q v R) and P v (Q ^ R) are equalivent to…

Answer 7

(P ^ Q) v (P ^ R) and (P v Q) ^ (P v R) respectively.

Question 8

P v ( P ^ Q) and P ^ (P v Q) are equal to…

Answer 8

P

Question 9

~(~P) is equal to…

Answer 9

P

Question 10

If Δ is defined as the symmetric difference operator, then A Δ B is equalivent to…

Answer 10

(A U B) \ (A n B)

or

(A \ B) U (B \ A)

Question 11

A \ B is equalivent to…

Answer 11

A \ B = {x |x ∈ A ^ x !∈ B}

Question 12

P → Q, is equalivent to…

Answer 12

~P v Q

or

~(P ^ ~Q)

Question 13

P ↔ Q is equalivent to…

Answer 13

(P → Q) ∧ (¬P → ¬Q)

or

(~P v Q) ^ (P v ~Q)

Question 14

A ⊆ B is equalivent to…

Answer 14

∀x(x ∈ A → x ∈ B)

Question 15

∀x P(x) means…

Answer 15

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.

Question 16

What does ∀x∈A,P(x) mean?

Answer 16

∀x(x∈A→P(x))

Question 17

∃x P(x) means…

Answer 17

There exists one value of x in the solution set of P(x), or that:

the solution set of P(x) does not equal ∅.

Question 18

What does ∃!xP(x) mean?

Answer 18

There exists exactly one x such that P(x) is true.

Question 19

What does ∃x∈A,P(x) mean?

Answer 19

∃x(x∈A∧P(x))

Question 20

¬∃xP(x) and ¬∀xP(x) are equalivent to…

Answer 20

¬∃xP(x)is equivalent to∀x¬P(x).
¬∀xP(x)is equivalent to∃x¬P(x).

Question 21

Suppose A = {1,2,3}, B = {4}, and C = ∅

If set F = {A, B,C}, does 1 ∈ F?

Answer 21

No.

1 is not in the set of F, but in the set of A.

Question 22

What is a power set? ℙ(A)

Suppose A is a set.

Answer 22

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}}

Question 23

∩F = ?

∪F = ?

Answer 23

∩F = {x | ∀A ∈ F(x ∈ A)} = {x | ∀A(A ∈ F → x ∈ A)}
∪F = {x | ∃A ∈ F(x ∈ A)} = {x | ∃A(A ∈ F ∧ x ∈ A)}