Connectives in George Tourlakis' Mathematical Logic

Question

In page 10 of Mathematical Logic, Tourlakis says that "Readers who have done some elementary course in logic, or in the context of a programming course, may have learned that ¬, ∨ are the only connectives one really needs since the rest can be expressed in terms of these two."
Capture from the book
How can we express the other connectives (→, ∧, ≡) using only ¬ and ∨?
Capture from the book

Hayden · Answer

The following logical equivalences (or equations in Boolean algebra) can be checked in a variety of ways:
begin{align}
P wedge Q & = neg (neg P vee neg Q) \
P to Q & = neg P vee Q \
P leftrightarrow Q & = (P to Q) wedge (Q to P) \
& = (neg P vee Q) wedge (neg Q vee P) \
& = negbigl( neg (neg P vee Q) vee neg (neg Q to P)bigr)
end{align}
The first, for example, can be considered as using De Morgan's Law, but could also be verified using truth tables.

Connectives in George Tourlakis' Mathematical Logic

One Answer

Add your own answers!

Ask a Question