Question

Because ⇒ is not commutative (that is, because p ⇒ q and q ⇒ p mean different things), it is not too surprising that ⇒ has neither an identity nor a zero. But there are a pair of related definitions that apply to this type of operator:

3.21 The left identity of a binary operator ◊ is a value iℓ such that, for any x, the expressions x and iℓ ◊ x are equivalent. The right identity of ◊ is a value ir such that, for any x, the expressions x and x ◊ ir are equivalent. (Again, some operators may not have left or right identities.) What are the left and right identities of ⇒ (if they exist)?

Answer 1

The truth table for the binary operator ⟹ is:

p	q	p⟹q
0	0	1
0	1	1
1	0	0
1	1	1

From the above table, we see that

• 1⟹0=0
• 1⟹1=1

Therefore, 1⟹x=x for any x.

Thus, 1 is the left identity of ⟹.

Also, from the above table, we see that

• 0⟹0=1
• 0⟹1=1

Therefore, 0⟹y=1 for any y, that is,

0⟹y ≠ 0 for all y.

Thus, there does not exist any y such that 0⟹y=0.

Hence, right identity of ⟹ does not exist.