Laws of Logic

Question 1

A ___ is a named commonly used equivalent statement.

Answer 1

Law of Logic

Question 2

Two “nots” cancel out

¬¬p ≡ p

Answer 2

Double Negation Law

Question 3

☐

Answer 3

Q.E.D

Question 4

For “or” and “and” the order of the two parts does not matter.

p∨q ≡ q∨p
p∧q ≡ q∧p

Answer 4

Commutative Law

Question 5

p ∨ (¬q ∧ r) ≡ (r ∧ ¬q) ∨ p
p ∨ (¬q ∧ r) ≡ p ∨ (r ∧ ¬q)
≡ (r ∧ ¬q) ∨ p

Which law was used in this proof?

Answer 5

Commutative Law

Question 6

For the “or” of 3 statements or the “and” of 3 statements the order in which the “or”s or the “and”s does not matter.

p ∨ (q ∨ r) ≡ (p ∨ q) ∨ r
p ∧ (q ∧ r) ≡ (p ∧ q) ∧ r

Answer 6

Associative Law

Question 7

Associative cannot be used on a mixed symbols. ___ does that.

p ∧ (q ∨ r) ̸≡ (p ∧ q) ∨ r

Answer 7

Distributive Law

Question 8

Associative cannot be used if a negation exists on the inner connective. ___ needs to be used first.

p ∨ ¬(q ∨ r) ̸≡ ¬(p ∨ q) ∨ r

Answer 8

DeMorgan’s

Question 9

p ∨ (¬q ∨ r) ≡ (p ∨ ¬q) ∨ r

By which law is this statement true?

Answer 9

Associative Law

Question 10

If we have an “and” on the outside of an “or”, then the “and” can be distributed into the “or”, and vice versa.

p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

Answer 10

Distributive Law

Question 11

Distributive cannot be used if a negation exists on the inner connective. ___ needs to be used first.

p ∨ ¬(q ∧ r) ̸≡ (p ∨ ¬q) ∧ (p ∨ ¬r)

Answer 11

DeMorgan’s

Question 12

A distribution rule for negations.

If we know that I did not “eat and swim”, then either “I did not eat” or “I did not swim”.

Negations, when distributed into a connective, invert the sign.
¬(p ∧ q) ≡ ¬p ∨ ¬q
¬(p ∨ q) ≡ ¬p ∧ ¬q

Answer 12

DeMorgan’s Law

Question 13

When using DeMorgan’s, do not remove the parenthesis if…

Answer 13

combined with other connectives

p ∧ ¬(q ∧ r) ̸≡ p ∧ ¬q ∨ ¬r
instead
p ∧ ¬(q ∧ r) ≡ p ∧ (¬q ∨ ¬r)

Question 14

Proof “¬(p ∧ (q ∨ r))” ≡ “¬p ∨ (¬q ∧ ¬r)”
¬(p ∧ (q ∨ r)) ≡ ¬p ∨ ¬(q ∨ r)
≡ ¬p ∨ (¬q ∧ ¬r)

Which law was used in this proof?

Answer 14

DeMorgan’s

Question 15

If a statement is “or” with itself and something else, we can ignore the something else.

p ∨ (p ∧ q) ≡ p
p ∧ (p ∨ q) ≡ p

Answer 15

Absorption Law

Question 16

p ∧ (p ∨ (p ∧ q)) ≡ p

By which law is this statement true?

Answer 16

Absorption Law

Question 17

If a statement or/and with itself is just the statement.

p ∨ p ≡ p
p ∧ p ≡ p

Answer 17

Idempotent Law

Question 18

(p ∧ q) ∨ (p ∧ q) ≡ p ∧ q

By which law is this statement true?

Answer 18

Idempotent Law

Question 19

The ___ law is like multiplying by 1 or adding 0. Applying the ___ means that the statement does not change. The ___ for True and False is not 1 or 0, but T and F.

p ∨ F ≡ p
p ∧ T ≡ p

Answer 19

Identity

Question 20

(p ∧ q) ∧ T ≡ p ∧ q

By which law is this statement true?

Answer 20

Identity Law

Question 21

Simple examples of Tautology or Contradiction.

p ∨ ¬p ≡ T
p ∧ ¬p ≡ F

Answer 21

Inverse Law

Question 22

p ∨ F ≡ p ∨ (¬q ∧ ¬¬q)

By which law is this statement true?

Answer 22

Inverse Law

Question 23

Occurs when a statement is removed by Tautologies or Contradictions.

p ∨ T ≡ T
p ∧ F ≡ F

Answer 23

Domination Law

Question 24

T ∨ F ≡ T

By which law is this statement true?

Answer 24

Domination Law

Question 25

Used to convert Implications into standard and/or/negation symbols. Since implication is True in 3 of 4 possible outcomes, it is similar to the “or” logic.

p → q ≡ ¬p ∨ q

Answer 25

Implication Identity

Question 26

Niether commutative or associative

Answer 26

Implication

Question 27

Proof “p → (q → r)” ≡ “¬p ∨ (¬q ∨ r)”
p → (q → r) ≡ ¬p ∨ (q → r)
≡ ¬p ∨ (¬q ∨ r)

Which law was used in this proof?

Answer 27

Implication Identity