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Oct 26 In-Class Exercise Thread.

Please post your solutions to the OCt 26 In-Class Exercise to this thread.

Best,

Chris

Please post your solutions to the OCt 26 In-Class Exercise to this thread. Best, Chris

-- Oct 26 In-Class Exercise Thread

R1 ((A⇒(B∧C))⇒D. (start)

R2 ¬(¬A∨(B∧C))∨ D (Implication elimination of R1)

R3 (A ∧ (¬B ∨ ¬C)) ∨ D (DeMorgan Law on R2)

R4 (A ∨ D) ∧ ((¬B ∨ ¬C)∨ D) (distributivity of ∨ or ∧ on R3)

(Edited: 2022-10-26)

R1 ((A⇒(B∧C))⇒D. (start) R2 ¬(¬A∨(B∧C))∨ D (Implication elimination of R1) R3 (A ∧ (¬B ∨ ¬C)) ∨ D (DeMorgan Law on R2) R4 (A ∨ D) ∧ ((¬B ∨ ¬C)∨ D) (distributivity of ∨ or ∧ on R3)

-- Oct 26 In-Class Exercise Thread

R1: ((A ⇒ (B ∧ C)) ⇒ D
R2: ¬ A ∨ (B ∧ C) ⇒ D (implication elimination)
R3: ¬ (¬ A ∨ (B ∧ C)) ∨ D (implication elimination)
R4: (A ∧ (¬ B ∨ ¬ C)) ∨ D (DeMorgan Law)

(Edited: 2022-10-26)

R1: ((A ⇒ (B &and; C)) ⇒ D <br> R2: ¬ A &or; (B &and; C) ⇒ D (implication elimination)<br> R3: ¬ (¬ A &or; (B &and; C)) &or; D (implication elimination)<br> R4: (A &and; (¬ B &or; ¬ C)) &or; D (DeMorgan Law)<br>

-- Oct 26 In-Class Exercise Thread

 ((A⇒(B∧C))⇒D
 1) (-A∨(B∧C))⇒D       implication elimination of A⇒(B∧C)
 2) -(-A∨(B∧C))∨D      implication elimination of (-A∨(B∧C))⇒D
 3) (A∧(-B∨-C))∨D      Demorgan
 4) (A∨D)∧((-B∨-C)∨D)  distributivity of ∨    
 4) (A∨D)∧(-B∨-C∨D)    associativity of ∨

(Edited: 2022-10-26)

((A⇒(B∧C))⇒D 1) (-A∨(B∧C))⇒D implication elimination of A⇒(B∧C) 2) -(-A∨(B∧C))∨D implication elimination of (-A∨(B∧C))⇒D 3) (A∧(-B∨-C))∨D Demorgan 4) (A∨D)∧((-B∨-C)∨D) distributivity of ∨ 4) (A∨D)∧(-B∨-C∨D) associativity of ∨

-- Oct 26 In-Class Exercise Thread

 R1: ((A ⇒ (B ∧ C)) ⇒ D
 R2: ¬(¬A ∨ (B ∧ C)) ∨ D
 R3: A ∧ ¬(B ∧ C) ∨ D
 R4: A ∧ ¬B ∨ ¬C ∨ D
 R5: A ∧ (¬B ∨ ¬C ∨ D)

R1: ((A ⇒ (B ∧ C)) ⇒ D R2: ¬(¬A ∨ (B ∧ C)) ∨ D R3: A ∧ ¬(B ∧ C) ∨ D R4: A ∧ ¬B ∨ ¬C ∨ D R5: A ∧ (¬B ∨ ¬C ∨ D)

-- Oct 26 In-Class Exercise Thread

A -> (B `∧ C)) -> D

By Implication Elimination:

`\neg`(`\neg`A V (B ∧ C)) V D

Distribute Negation:

(A ∧ (`\neg`B V `\neg`C)) V D

Distribute V:

((A ∧ `\neg`B) V (A ∧ `\neg`C)) V D

By DeMorgan:

(`\neg`A V B) ∧ (`\neg`A V C) ∧ `\neg`D

(Edited: 2022-10-26)

A -> (B @BT@∧ C)) -> D By Implication Elimination: @BT@\neg@BT@(@BT@\neg@BT@A V (B ∧ C)) V D Distribute Negation: (A ∧ (@BT@\neg@BT@B V @BT@\neg@BT@C)) V D Distribute V: ((A ∧ @BT@\neg@BT@B) V (A ∧ @BT@\neg@BT@C)) V D By DeMorgan: (@BT@\neg@BT@A V B) ∧ (@BT@\neg@BT@A V C) ∧ @BT@\neg@BT@D

-- Oct 26 In-Class Exercise Thread

1.). ((A⇒(B∧C))⇒D

2). ¬(¬A∨(B∧C))∨D

3). ¬((¬A∨B)∧(¬A∨C))∨D use[R∨(P∧Q) --> (P∨R)∧(Q∨R)]

4). (A∧¬B)∧(A∧¬C)∨D. use[¬¬A-->A]　and ¬(P∨Q) --> (¬P)∧(¬Q)

(Edited: 2022-10-26)

1.). ((A⇒(B∧C))⇒D 2). ¬(¬A∨(B∧C))∨D 3). ¬((¬A∨B)∧(¬A∨C))∨D use[R∨(P∧Q) --> (P∨R)∧(Q∨R)] 4). (A∧¬B)∧(A∧¬C)∨D. use[¬¬A-->A]　and ¬(P∨Q) --> (¬P)∧(¬Q)

-- Oct 26 In-Class Exercise Thread

Step 1: ((A => (B ∧ C)) => D Step 2: ((¬A ∨ (B ∧ C)) => D Step 3: ((¬A ∨ B) ∧ (¬A ∨ C) => D Step 4: ¬((¬A ∨ B) ∧ (¬A ∨ C) ∨ D Step 5: (A ∧ ¬B) ∨ (A ∧ ¬C) ∨ D Step 6: (A ∧ (¬B ∧ ¬C ∧ D))

-- Oct 26 In-Class Exercise Thread

 (A ⇒ (B ∧ C)) ⇒ D
 (¬A ∨ (B ∧ C)) ⇒ D                    Implication elimination
 ((¬A ∨ B) ∧ (¬A ∨ C)) ⇒ D             Distributivity of ∨
 ¬((¬A ∨ B) ∧ (¬A ∨ C)) ∨ D            Implication elimination
 (¬(¬A ∨ B) ∨ ¬(¬A ∨ C)) ∨ D           DeMorgan
 (A ∧ ¬B) ∨ (A ∧ ¬C) ∨ D               Demorgan
 A ∧ (¬B ∨ ¬C) ∨ D                     Distributivity of ∧

(A ⇒ (B ∧ C)) ⇒ D (¬A ∨ (B ∧ C)) ⇒ D Implication elimination ((¬A ∨ B) ∧ (¬A ∨ C)) ⇒ D Distributivity of ∨ ¬((¬A ∨ B) ∧ (¬A ∨ C)) ∨ D Implication elimination (¬(¬A ∨ B) ∨ ¬(¬A ∨ C)) ∨ D DeMorgan (A ∧ ¬B) ∨ (A ∧ ¬C) ∨ D Demorgan A ∧ (¬B ∨ ¬C) ∨ D Distributivity of ∧

-- Oct 26 In-Class Exercise Thread

 R1 (A => (B and C)) => D
 R2 not (not A or (B and C)) or D (implication elimination)
 R3 not ((not A or B) and (not A or C)) or D (distribution)
 R4 ((not A or B) and (not A or C)) and not D (DeMorgan)
 R5 ((not A or B) and not D) and ((not A or C) and not D) (distribution)
 R6 ((not A and not D) or (B and not D)) and ((not A and not D) or (C and not D))
 R7 ((A or D) or (not B or D)) and ((A or D) or (not C or D)) (DeMorgan)
 R8 (A or not B or D) and (A or not C or D) (associativity)

(Edited: 2022-10-26)

R1 (A => (B and C)) => D R2 not (not A or (B and C)) or D (implication elimination) R3 not ((not A or B) and (not A or C)) or D (distribution) R4 ((not A or B) and (not A or C)) and not D (DeMorgan) R5 ((not A or B) and not D) and ((not A or C) and not D) (distribution) R6 ((not A and not D) or (B and not D)) and ((not A and not D) or (C and not D)) R7 ((A or D) or (not B or D)) and ((A or D) or (not C or D)) (DeMorgan) R8 (A or not B or D) and (A or not C or D) (associativity)