# 5.2 Operations on Boolean values

The idea of a binary operation, and the use of infix notation, is not confined to numbers. Infix notation may be used for any process with two inputs from the same set. We look now at two binary operations on Boolean values that are often used.

The first of these takes two Boolean values, and returns **true** if *both* of the input values are **true** and returns **false** otherwise. It is a binary operation, and we shall write it using the infix notation ∧, where the symbol ∧ can be read as “and”. So *a* ∧ *b* is **true** only when *a* and *b* are both **true**. We can describe the function corresponding to this operation as below (The operation ∧ is also known as **conjunction**).

**function** (**infix**) (*a* ∧ *b* **on** *Bool*) **return in** *Bool*

**pre true**.

**post** The returned value is **true** if both *a* = **true** and *b* = **true**, and is **false** otherwise.

Alternatively, we could give the semantics of ∧ by listing the returned value in each of four cases, giving the four possible combinations of input values. Using this approach, we could express the postcondition as follows.

**post** The returned value is *c*, where

if

*a*=**true**and*b*=**true**then*c*=**true**if

*a*=**true**and*b*=**false**then*c*=**false**if

*a*=**false**and*b*=**true**then*c*=**false**if

*a*=**false**and*b*=**false**then*c*=**false**.

A second binary operation on Boolean values returns **true** if either (or both) of the two Boolean values is **true**. We write this operation using the infix symbol ∨ , where ∧ can be read as “or”. The corresponding function is described below. The operation ∨ is also known as **disjunction**.

**function** (**infix**) (*a* ∨ *b* **on** *Bool*) **return in** *Bool*

**pre true**.

**post** The returned value is **false** if both *a* = **false** and *b* = **false**, and is **true** otherwise.

## Activity 25

Give the semantics of ∨ by listing the returned value in each of four cases, giving the four possible combinations of input values.

### Discussion

We can express the postcondition in the description above as follows.

**post** The returned value is *c*, where

if

*a*=**true**and*b*=**true**then*c*=**true**if

*a*=**true**and*b*=**false**then*c*=**true**if

*a*=**false**and*b*=**true**then*c*=**true**if

*a*=**false**and*b*=**false**then*c*=**false**.

Another useful process on Boolean values inputs a single Boolean value, and returns the reverse of that value. This process, called **negation**, is not a binary operation, since it has one input only. We call this function *NOT*, and we have

NOT(false) =true, andNOT(true) =false.

## Activity 26

(a) Give a full description of the function

*NOT*.(b) If

*a*=**true**and*b*=**false**, evaluate each of:(i)

*a*∨*b*;(ii)

*NOT*(*a*∨*b*);(iii)

*a*∧ (*a*∨*b*).

### Discussion

(a) A description is given below.

**function** *NOT*(*a* **in** *Bool*) **return in** *Bool*

**pre true**.

**post** The returned value is **false** if *a* = **true** and is **true** if *a* = **false**.

(b) (i) Substituting for *a* and *b*, we have **true** ∨ **false** = **true**.

(ii) We should evaluate the term in brackets first: **true** ∨ **false** = **true**. Then *NOT*(**true**) = **false**. So *NOT*(*a* ∨ *b*) = **false**.

(iii) Again, evaluate the term in brackets first: (**true** ∨ **false**) = **true**. Then the given expression becomes **true** ∧ **true** which evaluates to **true**.