Use the truth tables method

Question

  1.  Use the truth tables method to determine whether (¬p v q)∧(q → ¬r ∧ ¬p)∧  (p v r) is satisfiable.
  2. Is the compound proposition ¬(P v Q v R) a contradiction? Justify.
  3. Prove using the laws of logic that A →  (B v C0=(A →  B) v( A →  C)
  4. Prove using the laws of logic that (¬Q → ¬p) →  (p → Q) = true
  5. Next prove using the laws of logic that (p →  r) ∧ (q →  r) = ((p v q) )→  r

Summary

Here we have given 5 questions. So We have to use the truth tables method to determine the given condition. And Next, we have to answer if a given condition is a contradiction. So that in the third, fourth, and fifth questions we have to prove the logic of the given condition.

Explanation

So here in the first question, we are using a truth table. In that truth table, a proposition is said to be satisfied if anyone one of its output/results is true.

P Q R ¬P ¬R ¬PvQ Q → R Q→  R ∧ ¬P (→ PVQ) ∧ Q→ R PVR Result
T T T F F T F F F T F
T T F F T T T F F T F
T F T F F F T F F T F
T F F F T T T F F T F
F T T T F T F F F T F
F T F T T T T T T F F
F F T T F F T T F T F
F F F T T T T T T T F

The given propositions are not a satisfaction as all the results are false.

For the given proposition contradiction is false. And it can be driven out of it. So, in the given condition

P Q R PV Q Pv Q v R (P v Q v R)
T T T T T F
T T F T T F
T F T T T F
T F F T T F
F T T T T F
F T F T T F
F F T F T F
F F F F F T

Here out of eight, seven outputs are false. As the majority is on the side of false. It can be said that the given condition is a contradiction.

We have to prove the given expression.

A→(B v C)

so that we can say that

A  → (B or C)

A→B or A→C

(A→B) V (A→C)

Next, we have to prove  (¬Q → ¬p) →  (p → Q) = true

(– Q → –P) → (P → Q)

–(Q → P) →  (P →  Q)

(P →  Q) →  ( P → Q)

=true

Hence proved

Here we have to prove (p →  r) ∧ (q →  r) = ((p v q) )→  r

(p →  r) ∧ (q →  r) So, here,

p implies to r and q implies to r

So here both p and q implies to r

(p v q) →  (r)

any one of p and q can imply to r

 

Also read, given pair of 16-bit binary data

 

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