Truth-tables-equivalent-statements-tautologies.pdf

TRUTH TABLES, EQUIVALENT STATEMENTS & TAUTOLOGIES

TRUTH TABLES

a.Construct a truth table for ~(~𝓹 ⋁ 𝓺) ⋁ 𝓺. b.Use the truth table from part a to determine the truth value of ~(~𝓹 ⋁ 𝓺) ⋁ 𝓺, given that 𝓹 is true and 𝓺 is false.

~(~𝓹 ⋁ 𝓺) ⋁ 𝓺 𝓹

𝓺

~(~𝓹 ⋁ 𝓺) ⋁ 𝓺 𝓹

𝓺

~𝓹

~(~𝓹 ⋁ 𝓺) ⋁ 𝓺 𝓹

𝓺

~𝓹 ~𝓹 ⋁ 𝓺

~(~𝓹 ⋁ 𝓺) ⋁ 𝓺 𝓹

𝓺

~𝓹 ~𝓹 ⋁ 𝓺

~(~𝓹 ⋁ 𝓺)

~(~𝓹 ⋁ 𝓺) ⋁ 𝓺 𝓹

𝓺

~𝓹 ~𝓹 ⋁ 𝓺

~(~𝓹 ⋁ 𝓺)

~(~𝓹 ⋁ 𝓺) ⋁ 𝓺

~(~𝓹 ⋁ 𝓺) ⋁ 𝓺 𝓹

𝓺

~𝓹 ~𝓹 ⋁ 𝓺

T T F F

T F T F

F F T T

T F T T

~(~𝓹 ⋁ 𝓺)

~(~𝓹 ⋁ 𝓺) ⋁ 𝓺

F T F F

T T T F

~(~𝓹 ⋁ 𝓺) ⋁ 𝓺 FOR 𝓹 IS TRUE AND 𝓺 IS FALSE 𝓹

𝓺

~𝓹 ~𝓹 ⋁ 𝓺

T T F F

T F T F

F F T T

T F T T

~(~𝓹 ⋁ 𝓺)

~(~𝓹 ⋁ 𝓺) ⋁ 𝓺

F T F F

T T T F

TRUTH TABLES

a.Construct a truth table for (𝓹 ⋀ 𝓺) ⋀ (~𝓻 ⋁ 𝓺). b.Use the truth table from part a to determine the truth value of (𝓹 ⋀ 𝓺) ⋀ (~𝓻 ⋁ 𝓺), given that 𝓹 is true, 𝓺 is true, and 𝓻 is false.

(𝓹 ⋀ 𝓺) ⋀ (~𝓻 ⋁ 𝓺) 𝓹

𝓺

𝓻

(𝓹 ⋀ 𝓺) ⋀ (~𝓻 ⋁ 𝓺) 𝓹

𝓺

𝓻

𝓹⋀𝓺

(𝓹 ⋀ 𝓺) ⋀ (~𝓻 ⋁ 𝓺) 𝓹

𝓺

𝓻

𝓹 ⋀ 𝓺 ~𝓻

(𝓹 ⋀ 𝓺) ⋀ (~𝓻 ⋁ 𝓺) 𝓹

𝓺

𝓻

𝓹 ⋀ 𝓺 ~𝓻 ~𝓻 ⋁ 𝓺

(𝓹 ⋀ 𝓺) ⋀ (~𝓻 ⋁ 𝓺) 𝓹

𝓺

𝓻

𝓹 ⋀ 𝓺 ~𝓻 ~𝓻 ⋁ 𝓺 (𝓹 ⋀ 𝓺) ⋀ (~𝓻 ⋁ 𝓺)

𝓹

𝓺

𝓻

𝓹 ⋀ 𝓺 ~𝓻 ~𝓻 ⋁ 𝓺 (𝓹 ⋀ 𝓺) ⋀ (~𝓻 ⋁ 𝓺)

(𝓹 ⋀ 𝓺) ⋀ (~𝓻 ⋁ 𝓺)

𝓹

𝓺

𝓻

𝓹 ⋀ 𝓺 ~𝓻 ~𝓻 ⋁ 𝓺 (𝓹 ⋀ 𝓺) ⋀ (~𝓻 ⋁ 𝓺)

T ⋁ 𝓺)T (𝓹T⋀ 𝓺)T⋀ (~𝓻 T T T F F F F

T F F T T F F

F T F T F T F

T F F F F F F

F T F T F T F T

T T F T T T F T

T T F F F F F F

𝓹

𝓺

𝓻

𝓹 ⋀ 𝓺 ~𝓻 ~𝓻 ⋁ 𝓺 (𝓹 ⋀ 𝓺) ⋀ (~𝓻 ⋁ 𝓺)

T ⋁ 𝓺)T (𝓹T⋀ 𝓺)T⋀ (~𝓻 T T T F F F F

T F F T T F F

F T F T F T F

T F F F F F F

F T F T F T F T

T T F T T T F T

T T F F F F F F

TRUTH TABLES

Construct a truth table for 𝓹 ⋁ ~(𝓹 ⋀ ~𝓺)

THREE-VALUESD LOGIC

• A statement is true, false, or “somewhere between true and false.” • true (T), false (F), or maybe(M)

𝓹

𝓺

~𝓹

𝓹⋀𝓺

𝓹⋁ 𝓺

𝓹 T T T M M M F F F

𝓺 T M F T M F T M F

~𝓹 F F F M M M T T T

𝓹⋀𝓺 T M F M M F F F F

𝓹⋁𝓺 T T T T M M T M F

TRUTH TABLES

Construct a truth table for • ~(𝓹 ⋁ ~𝓺) • ~𝓹 ⋀ 𝓺

EQUIVALENT STATEMENTS Two statements are equivalent if they both have the same truth value for all possible truth values of their simple statements. Equivalent statements have identical truth values in the final columns of their truth tables. The notation 𝓹 ≡ 𝓺 is used to indicate that the statements 𝓹 and 𝓺 are equivalent.

DE MORGAN’S LAWS FOR STATEMENTS

• For any statements 𝓹 and 𝓺, ~(𝓹 ⋁ 𝓺) ≡ ~𝓹 ⋀ ~𝓺 ~(𝓹 ⋀ 𝓺) ≡ ~𝓹 ⋁ ~𝓺 • it can be used to restate certain English sentences in an equivalent form.

TAUTOLOGIES AND SELF-CONTRADICTIONS

• TAUTOLOGIES A statement that is always true • SELF-CONTRADICTIONS A statement that is always false

TRUTH TABLES

Construct a truth table for • 𝓹 ⋁ (~𝓹 ⋁ 𝓺)

CONDITIONAL AND BICONDITIONAL

• CONDITIONAL STATEMENTS can be written in form if 𝓹, then 𝓺 or if 𝓹, 𝓺 form. statement 𝓹, antecedent statement 𝓺, consequent

TRUTH VALUE FOR 𝓹 → 𝓺

𝓹: You can use word processor. 𝓺: You can create a webpage. Antecedent (T), Consequent (T) Truth value (T)

TRUTH VALUE FOR 𝓹 → 𝓺

𝓹: You can use word processor. 𝓺: You can create a webpage. Antecedent (T), Consequent (F) Truth value (F)

TRUTH VALUE FOR 𝓹 → 𝓺

𝓹: You can use word processor. 𝓺: You can create a webpage. Antecedent (F), Consequent (T) Truth value (T)

TRUTH VALUE FOR 𝓹 → 𝓺

𝓹: You can use word processor. 𝓺: You can create a webpage. Antecedent (F), Consequent (F) Truth value (T)

TRUTH TABLE FOR 𝓹 → 𝓺

𝓹

𝓺

𝓹→𝓺

T

T

T

T

F

F

F

T

T

F

F

T

FIND THE TRUTH VALUE OF A CONDITIONAL

a. If 2 is an integer, then 2 is a rational number. b. If 3 is a negative number, then 5 > 7. c. If 5 > 3, then 2 + 7 = 4

TRUTH TABLES

Construct a truth table for • [𝓹 ⋀ (𝓺 ⋁ ~𝓹)] → ~𝓹

TRUTH TABLES

Construct a truth table for • ~𝓹 ⋁ 𝓺 • 𝓹 → 𝓺 ≡ ~𝓹 ⋁ 𝓺

WRITE A CONDITIONAL IN ITS EQUIVALENT DISJUNCTIVE FORM

a. If I could play the guitar, I would join the band. b. If Cam Newton cannot play, then his team will lose.

TRUTH TABLES

Construct a truth table for • ~(𝓹 → 𝓺) ≡ 𝓹 ⋀ ~𝓺

WRITE THE NEGATION OF A CONDITIONAL

a. If they pay me the money, I will sign the contract. b. If the lines are parallel, then they do not intersect.

CONDITIONAL AND BICONDITIONAL

• BICONDITIONAL STATEMENTS the statement (𝓹 → 𝓺) ⋀ (𝓺 → 𝓹) denoted by 𝓹 ↔ 𝓺. • 𝓹 ↔ 𝓺 ≡ (𝓹 → 𝓺) ⋀ (𝓺 → 𝓹)

WRITE SYMBOLIC BICONDITIONAL STATEMENTS IN WORDS

Let 𝓹, 𝓺, and 𝓻 represent the following: 𝓹: She will go on vacation. 𝓺: She cannot take the train. 𝓻: She cannot get a loan. a. 𝓹 ↔ ~𝓺

b. ~𝓻 ↔ ~𝓹

TRUTH TABLE FOR 𝓹 ↔ 𝓺

𝓹

𝓺

𝓹↔𝓺

T

T

T

T

F

F

F

T

F

F

F

T

DETERMINE THE TRUTH VALUE OF A BICONDITIONAL

a. 𝑥 + 4 = 7 if and only if x = 3. 2 b. 𝑥 = 36 if and only if x = 6.

DETERMINE THE TRUTH VALUE OF A BICONDITIONAL

a. 𝑥 + 4 = 7 if and only if x = 3. 2 b. 𝑥 = 36 if and only if x = 6.