# Thread: What is called this logical formula?

1. ## What is called this logical formula?

Hello,

I want to know how the following formula in logic is called:
$\displaystyle (A\Rightarrow B)\Rightarrow ((A\wedge B)\Leftrightarrow A)$

Remarks:
(1) I once read in a book ("Theory and Problems of Logic", McGraw-Hill) calling it "the absorption law".

(2) In many textbooks, I see:
"the absorption law"----$\displaystyle (A\wedge (A\vee B))\Leftrightarrow A$
"the idempotent law"----$\displaystyle (A\wedge A)\Leftrightarrow A$

(3) The formula in question is often found in set theory in the form:
$\displaystyle A\subset B\Rightarrow A\cap B = A$

2. Originally Posted by wisterville
Hello,

I want to know how the following formula in logic is called:
$\displaystyle (A\Rightarrow B)\Rightarrow ((A\wedge B)\Leftrightarrow A)$

Remarks:
(1) I once read in a book ("Theory and Problems of Logic", McGraw-Hill) calling it "the absorption law".

(2) In many textbooks, I see:
"the absorption law"----$\displaystyle (A\wedge (A\vee B))\Leftrightarrow A$
"the idempotent law"----$\displaystyle (A\wedge A)\Leftrightarrow A$

(3) The formula in question is often found in set theory in the form:
$\displaystyle A\subset B\Rightarrow A\cap B = A$

It's a strengthened version of the following (usually derived) rule of inference, known as "absorption" in some texts.

A -> B |- A -> (A ^ B)

Clearly, strengthening the conclusion provides nothing new, since A ^ B |- A.
(E.g., in an NDS this is just the primitive rule, caret elimination.)

3. Thank you.

I was confused because I saw the term "absorption " used in another context.

Maybe it is viewed as a special version of
"the absorption law"----$\displaystyle (A\wedge(A\vee B))\Leftrightarrow A$
if we take into account that
$\displaystyle A\Rightarrow B\vdash (A\vee B)\Leftrightarrow B$.