Logic (slides)

Propositions and Logical OperationsDefinition: A statement or proposition is a declarative sentence that is either true (T) or false (F), but not both.

Example: Which of the following are statements?

a. It is raining.

b. 2 + 3 = 5

c. Do you speak English?

d. 3 − 𝑥𝑥 = 5

e. Take two aspirins.© S. Turaev, CSC 1700 Discrete Mathematics 2

Propositions and Logical Operations The letters 𝑝𝑝, 𝑞𝑞, 𝑟𝑟 are denote propositional variables

• 𝑝𝑝: I am teaching.

• 𝑞𝑞: 3 × 23 = 70

Compound statements: propositional variables combined by logical connectives (and, or, if … then, …):

• 𝑝𝑝 and 𝑞𝑞.

• 𝑝𝑝 or 𝑞𝑞.

• If 𝑝𝑝 then 𝑞𝑞.

3© S. Turaev, CSC 1700 Discrete Mathematics

Propositions and Logical OperationsDefinition: If 𝑝𝑝 is a statement, the negation of 𝑝𝑝 is the statement not 𝑝𝑝, denoted by ~𝑝𝑝 (sometimes, ¬𝑝𝑝, �̅�𝑝).

~𝑝𝑝: “It is not the case that 𝑝𝑝”.

Example:

𝑝𝑝: 2 + 3 > 1 ~𝑝𝑝:

𝑞𝑞: It is cold. ~𝑝𝑝:


𝑝𝑝 ~𝑝𝑝T FF T

Propositions and Logical OperationsDefinition: If 𝑝𝑝 and 𝑞𝑞 are statements, the conjunction of 𝑝𝑝and 𝑞𝑞 is the compound statement 𝑝𝑝 and 𝑞𝑞, denoted by 𝑝𝑝 ∧ 𝑞𝑞.

Example:

𝑝𝑝: It is raining. 𝑞𝑞: It is cold.

𝑝𝑝: 2 < 3. 𝑞𝑞: −3 < −2.

𝑝𝑝 ∧ 𝑞𝑞:


𝑝𝑝 𝑞𝑞 𝑝𝑝 ∧ 𝑞𝑞T T TT F FF T FF F F

Propositions and Logical OperationsDefinition: If 𝑝𝑝 and 𝑞𝑞 are statements, the disjunction of 𝑝𝑝and 𝑞𝑞 is the compound statement 𝑝𝑝 or 𝑞𝑞, denoted by 𝑝𝑝 ∨𝑞𝑞.

Example:

𝑝𝑝: It is raining. 𝑞𝑞: It is cold.

𝑝𝑝: 2 < 3. 𝑞𝑞: −3 < −2.

𝑝𝑝 ∨ 𝑞𝑞:


𝑝𝑝 𝑞𝑞 𝑝𝑝 ∧ 𝑞𝑞T T TT F TF T TF F F

Propositions and Logical OperationsA compound statement may have many components:

𝑝𝑝 ∨ (𝑞𝑞 ∧ (~ 𝑝𝑝 ∧ 𝑟𝑟 ))

Example: Make a truth table for 𝑝𝑝 ∧ 𝑞𝑞 ∨ ~𝑝𝑝.


𝑝𝑝 𝑞𝑞 𝑝𝑝 ∧ 𝑞𝑞 ~𝑝𝑝 ∨T TT FF TF F

Propositions and Logical OperationsDefinition: If 𝑝𝑝 and 𝑞𝑞 are statements, the compound statement “if 𝑝𝑝 then 𝑞𝑞”, denoted 𝑝𝑝 ⇒ 𝑞𝑞, is called a conditional statement or implication.

The statement 𝑝𝑝 is called antecedent or hypothesis and 𝑞𝑞is called the consequent or conclusion.

Example:

𝑝𝑝: I am hungry. 𝑞𝑞: I will eat.

𝑝𝑝: It is snowing. 𝑞𝑞: 3 + 2 = 5.

𝑝𝑝 ⇒ 𝑞𝑞:


𝑝𝑝 𝑞𝑞 𝑝𝑝 ⇒ 𝑞𝑞T T TT F FF T TF F T

Propositions and Logical OperationsDefinition: If 𝑝𝑝 ⇒ 𝑞𝑞 is an implication, then

the converse of 𝑝𝑝 ⇒ 𝑞𝑞 is the implication 𝑞𝑞 ⇒ 𝑝𝑝

the inverse of 𝑝𝑝 ⇒ 𝑞𝑞 is the implication ~𝑝𝑝 ⇒ ~𝑞𝑞

the contrapositive of 𝑝𝑝 ⇒ 𝑞𝑞 is the implication ~𝑞𝑞 ⇒~𝑝𝑝

Example: 𝑝𝑝 ⇒ 𝑞𝑞: “If it is raining then I get wet” then

𝑞𝑞 ⇒ 𝑝𝑝, ~𝑝𝑝 ⇒ ~𝑞𝑞, ~𝑞𝑞 ⇒ ~𝑝𝑝?


Propositions and Logical OperationsDefinition: If 𝑝𝑝 and 𝑞𝑞 are statements, the compound statement “𝑝𝑝 if and only if 𝑞𝑞”, denoted 𝑝𝑝 ⇔ 𝑞𝑞, is called an equivalence or biconditional.

Example:

𝑝𝑝: 3 > 2. 𝑞𝑞: 0 < 3 − 2.

𝑝𝑝: It is snowing. 𝑞𝑞: 3 + 2 = 5.

𝑝𝑝 ⇔ 𝑞𝑞:


𝑝𝑝 𝑞𝑞 𝑝𝑝 ⇒ 𝑞𝑞T T TT F FF T FF F T

Propositions and Logical OperationsExample: Compute the truth table of the statement

𝑝𝑝 ⇒ 𝑞𝑞 ⇔ (~𝑞𝑞 ⇒ ~𝑝𝑝)


𝑝𝑝 𝑞𝑞 𝑝𝑝 ⇒ 𝑞𝑞 ~𝑞𝑞 ~𝑝𝑝 ~𝑞𝑞 ⇒ ~𝑝𝑝 ⇔T TT FF TF F

Propositions and Logical OperationsDefinition: A statement that is true for all possible values of its propositional variables is called a tautology.

Definition: A statement that is false for all possible values of its propositional variables is called a contradiction or an absurdity.

Definition: A statement that can be either true or falsefor all possible values of its propositional variables is called contingency.


Propositions and Logical OperationsExample:

The statement 𝑝𝑝 ∨ ~𝑝𝑝:

The statement 𝑝𝑝 ∧ ~𝑝𝑝:

The statement 𝑝𝑝 ⇒ 𝑞𝑞 ∧ (𝑝𝑝 ∨ 𝑞𝑞):


Propositions and Logical OperationsDefinition: We say that the statements 𝑝𝑝 and 𝑞𝑞 are logically equivalent (or simply equivalent), denoted by 𝑝𝑝 ≡ 𝑞𝑞, if 𝑝𝑝 ⇔ 𝑞𝑞 is tautology.

Example: Show that

𝑝𝑝 ∨ 𝑞𝑞 ≡ 𝑞𝑞 ∨ 𝑝𝑝

𝑝𝑝 ⇒ 𝑞𝑞 ≡ ~𝑝𝑝 ∨ 𝑞𝑞


Propositions and Logical OperationsDefinition: A predicate or a propositional function is a noun/verb phrase template that describes a property of objects, or a relationship among objects represented by the variables:

Example: 𝑃𝑃 𝑥𝑥 : “𝑥𝑥 is integer less than 8.”

𝑃𝑃 1 =

𝑃𝑃 10 =

𝑃𝑃 −11 =


Propositions and Logical OperationsDefinition: The universal quantification of a predicate 𝑃𝑃 𝑥𝑥 is the statement “For all values of 𝑥𝑥 (for every 𝑥𝑥, for each 𝑥𝑥, for any 𝑥𝑥), 𝑃𝑃 𝑥𝑥 is true” and is denoted by ∀𝑥𝑥𝑃𝑃 𝑥𝑥 .

Example: 𝑃𝑃 𝑥𝑥 : “− −𝑥𝑥 = 𝑥𝑥” is a predicate that is true for all real numbers.

∀𝑥𝑥𝑃𝑃 𝑥𝑥 =

Example: 𝑄𝑄 𝑥𝑥 : “𝑥𝑥 + 1 < 4”.

∀𝑥𝑥𝑄𝑄 𝑥𝑥 =


Propositions and Logical OperationsA predicate may contain several variables.

Example: 𝑄𝑄 𝑥𝑥, 𝑦𝑦 : 𝑥𝑥 + 𝑦𝑦 = 𝑦𝑦 + 𝑥𝑥

∀𝑥𝑥∀𝑦𝑦𝑄𝑄 𝑥𝑥,𝑦𝑦 =

Example: Write the following statement in the form of a predicate and quantifier:

“The sum of any two integers is even number.”


Propositions and Logical OperationsDefinition: The existential quantification of a predicate 𝑃𝑃 𝑥𝑥 is the statement “There exists a value of 𝑥𝑥, for which 𝑃𝑃 𝑥𝑥 is true” and is denoted by ∃𝑥𝑥𝑃𝑃 𝑥𝑥 .

Example: 𝑃𝑃 𝑥𝑥 : “−𝑥𝑥 = 𝑥𝑥”.

∃𝑥𝑥𝑃𝑃 𝑥𝑥 =

Example: 𝑄𝑄 𝑥𝑥 : “𝑥𝑥 + 1 < 4”.

∃𝑥𝑥𝑄𝑄 𝑥𝑥 =


Algebraic PropertiesCommutative properties:

𝑝𝑝 ∨ 𝑞𝑞 ≡ 𝑞𝑞 ∨ 𝑝𝑝 𝑝𝑝 ∧ 𝑞𝑞 ≡ 𝑞𝑞 ∧ 𝑝𝑝

Associative properties:

𝑝𝑝 ∨ 𝑞𝑞 ∨ 𝑟𝑟 ≡ 𝑝𝑝 ∨ 𝑞𝑞 ∨ 𝑟𝑟 𝑝𝑝 ∧ 𝑞𝑞 ∧ 𝑟𝑟 ≡ 𝑝𝑝 ∧ 𝑞𝑞 ∧ 𝑟𝑟

Distributive properties:

𝑝𝑝 ∨ 𝑞𝑞 ∧ 𝑟𝑟 ≡ 𝑝𝑝 ∨ 𝑞𝑞 ∧ 𝑝𝑝 ∨ 𝑟𝑟 𝑝𝑝 ∧ 𝑞𝑞 ∨ 𝑟𝑟 ≡ 𝑝𝑝 ∧ 𝑞𝑞 ∨ 𝑝𝑝 ∧ 𝑟𝑟


Algebraic PropertiesIdempotent properties:

𝑝𝑝 ∨ 𝑝𝑝 ≡ 𝑝𝑝

𝑝𝑝 ∧ 𝑝𝑝 ≡ 𝑝𝑝

Properties of negation:

~(~𝑝𝑝) ≡ 𝑝𝑝

~ 𝑝𝑝 ∨ 𝑞𝑞 ≡ ~𝑝𝑝 ∧ ~𝑞𝑞

~ 𝑝𝑝 ∧ 𝑞𝑞 ≡ ~𝑝𝑝 ∨ ~𝑞𝑞


Algebraic PropertiesProperties of implication:

𝑝𝑝 ⇒ 𝑞𝑞 ≡ ~𝑝𝑝 ∨ 𝑞𝑞

𝑝𝑝 ⇒ 𝑞𝑞 ≡ ~𝑞𝑞 ⇒ ~𝑝𝑝

𝑝𝑝 ⇔ 𝑞𝑞 ≡ 𝑝𝑝 ⇒ 𝑞𝑞 ∧ 𝑞𝑞 ⇒ 𝑝𝑝

~ 𝑝𝑝 ⇒ 𝑞𝑞 ≡ 𝑝𝑝 ⇒ ~𝑞𝑞

~ 𝑝𝑝 ⇔ 𝑞𝑞 ≡ 𝑝𝑝 ∧ ~𝑞𝑞 ∨ 𝑞𝑞 ∧ ~𝑝𝑝


Algebraic PropertiesProperties of quantifiers:

~ ∀𝑥𝑥𝑃𝑃 𝑥𝑥 ≡ ∃𝑥𝑥~𝑃𝑃 𝑥𝑥

~ ∃𝑥𝑥𝑃𝑃 𝑥𝑥 ≡ ∀𝑥𝑥~𝑃𝑃 𝑥𝑥

∃𝑥𝑥 𝑃𝑃 𝑥𝑥 ⇒ 𝑄𝑄 𝑥𝑥 ≡ ∀𝑥𝑥𝑃𝑃 𝑥𝑥 ⇒ ∃𝑥𝑥𝑄𝑄 𝑥𝑥

∃𝑥𝑥 𝑃𝑃 𝑥𝑥 ∨ 𝑄𝑄 𝑥𝑥 ≡ ∃𝑥𝑥𝑃𝑃 𝑥𝑥 ∨ ∃𝑥𝑥𝑄𝑄 𝑥𝑥

∀𝑥𝑥 𝑃𝑃 𝑥𝑥 ∧ 𝑄𝑄 𝑥𝑥 ≡ ∀𝑥𝑥𝑃𝑃 𝑥𝑥 ∧ ∀𝑥𝑥𝑄𝑄 𝑥𝑥

∀𝑥𝑥𝑃𝑃 𝑥𝑥 ∨ ∀𝑥𝑥𝑄𝑄 𝑥𝑥 ⇒ ∀𝑥𝑥 𝑃𝑃 𝑥𝑥 ∨ 𝑄𝑄 𝑥𝑥

∃𝑥𝑥 𝑃𝑃 𝑥𝑥 ∧ 𝑄𝑄 𝑥𝑥 ⇒ ∃𝑥𝑥𝑃𝑃 𝑥𝑥 ∨ ∃𝑥𝑥𝑄𝑄 𝑥𝑥22© S. Turaev, CSC 1700 Discrete Mathematics

Algebraic PropertiesTautologies:

𝑝𝑝 ⇒ 𝑞𝑞 ∧ 𝑞𝑞 ⇒ 𝑟𝑟 ⇒ (𝑝𝑝 ⇒ 𝑟𝑟)23© S. Turaev, CSC 1700 Discrete Mathematics

𝑝𝑝 ∧ 𝑞𝑞 ⇒ 𝑝𝑝

𝑝𝑝 ⇒ 𝑝𝑝 ∨ 𝑞𝑞

~𝑝𝑝 ⇒ 𝑝𝑝 ⇒ 𝑞𝑞

𝑝𝑝 ∧ 𝑝𝑝 ⇒ 𝑞𝑞 ⇒ 𝑞𝑞

~𝑞𝑞 ∧ 𝑝𝑝 ⇒ 𝑞𝑞 ⇒ ~𝑝𝑝

𝑝𝑝 ∧ 𝑞𝑞 ⇒ 𝑞𝑞

𝑞𝑞 ⇒ 𝑝𝑝 ∧ 𝑞𝑞

~ 𝑝𝑝 ⇒ 𝑞𝑞 ⇒ 𝑝𝑝

~𝑝𝑝 ∧ 𝑝𝑝 ∨ 𝑞𝑞 ⇒ 𝑞𝑞

Logic (slides)

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