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1. Simplify the symbolic statements.

(b) (p≥7)∧(p<12), 7≤p<12

(c) (x>5)∧(x<7), 5<x<7

(d) (x<4)∧(x<6), x<4

(e) $(y<4)∧(y^2 <9), y^2<9$

(f) (x≥0)∧(x≤0), x=0

2. Express each of your simplified statements from question 1 in natural English.

(b) p is greater than or equal to 7 and less than 12.

(c) x is greater than 5, and less than 7.

(d) x is less than 4

(e) y squared is less than 9

(f) x is equal to 0

3. What strategy would you adopt to show that the conjunction φ1 ∧ φ2 ∧ . . . ∧ φn is true?

show that all of φ1 , φ2 , . . . , φn are true

4. What strategy would you adopt to show that the conjunction φ1 ∧ φ2 ∧ . . . ∧ φn is false?

show that one of φ1 , φ2 , . . . , φn is false.

5. Simplify the symbolic statements

(a) (π>3)∨(π>10), π>3 (b) (x<0)∨(x>0), x≠0 (c) (x=0)∨(x>0), x≥0 (d) (x>0)∨(x≥0), x≥0 (e) (x>3)∨($x^2$ >9), $x^2$ >9

6. Express the above symbolic statements in natural English.

(a) π is greater than 3

(b) x is not equal to 0

(c) x is equal to or greater than 0

(d) x is equal to or greater than 0

(e) x squared is greater than 9

7. What strategy would you adopt to show that the disjunction φ1 ∨ φ2 ∨ . . . ∨ φn is true?

show that one of φ1 , φ2 , . . . , φn is true

8. What strategy would you adopt to show that the disjunction φ1 ∨ φ2 ∨ . . . ∨ φn is false?

show that all φ1 , φ2 , . . . , φn are false

9. Simplify the symbolic statements

(a) ¬(π > 3.2), π ≤ 3.2 (b) ¬(x < 0), x ≥ 0 (c) ¬($x^2$ > 0), x = 0 (d) ¬(x = 1), x≠1 (e) ¬¬ψ, ψ

10. Express the above symbolic statements in natural English.

(a) negation of / not the case that Pai is greater than 3.2, Pai is less than and equal to 3.2 (b) negation of / not the case that x is less than 0, x is greater than and equal to 0 (c) negation of / not the case that x squared is greater than 0, x is equal to 0 (d) negation of / not the case that x equal to 1, x not equal to 1 (e) double negation of / not the case that ψ(Psi), ψ

11. Express in logical notation

(a) D ∧ Y

(b) ¬Y ∧ T ∧ D

(c) ¬(Y ∧ D)

(d) T ∧ ¬Y ∧ ¬D

(e) ¬T ∧ Y ∧ D

DISCUSS

Guilty and Proven

Truth Table, Binary logic, True or False

G P G ∧ P   description
T T T ✔︎ Correctly Proven Guilty
T F F Incorrectly Proven Innocent
F T F Incorrectly Proven Guilty
F F F ✔︎ Correctly Proven Innocent

Not disPleased

Not disPleased is neutral, this is not a binary logic but a triary logic

triary logic

Positive neutral Negative
Pleased not displeased disPleased
Pleased ¬(¬ pleased)  
¬ pleased    
     

¬ pleased = displeased

¬ displeasedpleased

¬ pleased < ¬ ( ¬ pleased) < pleased

displeased < ¬ displeased < pleased