Question Number 1656 by Rasheed Soomro last updated on 29/Aug/15

Commented by Yozzian last updated on 29/Aug/15

Commented by 112358 last updated on 29/Aug/15

Commented by Yozzy last updated on 29/Aug/15

Commented by Rasheed Soomro last updated on 30/Aug/15

Commented by Yozzy last updated on 30/Aug/15
![Propose that p=A⇒B,q=B⇒C and r=A⇒C . To prove there being a transitive property of implication is like saying (I think) to show that [(A⇒B)∧(B⇒C)]⇒(A⇒C)=T, i.e (p∧q)⇒r is a tautology. Without truth tables, I′ll attempt to use the laws of Boolean algebra to simplify the logic expression to hopefully end up having a tautology. Note that μ⇒z≡∽μ∨z. [(A⇒B)∧(B⇒C)]⇒(A⇒C)≡[(∽A∨B)∧(∽B∨C)]⇒(∽A∨C) lhs≡∽[(∽A∨B)∧(∽B∨C)]∨(∽A∨C) lhs≡[∽(∽A∨B)∨∽(∽B∨C)]∨(∽A∨C) De Morgan′s law lhs≡[(A∧∽B)∨(B∧∽C)]∨(∽A∨C) De Morgan′s law, Double negation law lhs≡[∽A∨(A∧∽B)]∨[C∨(B∧∽C)] Commutative law,Associative law lhs≡[(∽A∨A)∧(∽A∨∽B)]∨[(C∨B)∧(C∨∽C)] Distribution law lhs≡[T∧(∽A∨∽B)]∨[(C∨B)∧T] Complementary law lhs≡(∽A∨∽B)∨(C∨B) Identity law lhs≡(∽B∨B)∨(C∨∽A) Commutative law,Associative law lhs≡T∨(C∨∽A) Complementary law lhs≡T Domination law The simplification of the Boolean statement has given us a constant truth value of T, so the statement is perpetually true. Hence,a transitive property of the implication logic as been proven. ■](https://www.tinkutara.com/question/Q1662.png)
Commented by Yozzy last updated on 30/Aug/15
![A∣B∣C∣A⇒B∣B⇒C∣A⇒C∣(A⇒B)∧(B⇒C)∣[(A⇒B)∧(B⇒C)]⇒(A⇒C) 0 0 0 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 1 1 1 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 1 0 1 1 1 0 1 0 0 0 1 1 1 1 1 1 1 1 1 The last column to the right indicates that the statement [(A⇒B)∧(B⇒C)]⇒(A⇒C) is always true. Therefore, this proves the existance of a transitive property of the implication logic. (I′ll check over my Boolean simplification.Checked.)](https://www.tinkutara.com/question/Q1663.png)
Commented by Rasheed Soomro last updated on 30/Aug/15

Commented by Yozzy last updated on 30/Aug/15

Commented by Rasheed Soomro last updated on 30/Aug/15

Commented by 112358 last updated on 30/Aug/15

Commented by 123456 last updated on 30/Aug/15

Commented by Rasheed Ahmad last updated on 30/Aug/15
