Conjunction

What is a conjunction?

A conjunction is a binary operation in the sense that it requires two operands. When two propositions are connected using a conjunction operation, it becomes a new proposition whose value depends on the truth values of the operands. Another name for the conjunction operator is the AND operator.

Example

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Consider the following propositions:

• $D$: Joseph is taking a discrete mathematics course.
• $C$: Joseph is taking a calculus course.

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Let’s make a new proposition $J$ by taking the conjunction of $D$ and $C$.

• $J$: “Joseph is taking a discrete mathematics course” and “Joseph is taking a calculus course.”

We can simplify $J$ without changing the meaning.

• $J$: Joseph is taking both discrete mathematics and calculus courses.

We use the $\land$ symbol to represent the conjunction operation. So,

$$J \equiv D \land C.$$

The truth value of $J$ is true only when $D$ and $C$ both are true and false otherwise.

It is intuitive to think of conjunction as an “and” in everyday language. In fact, that is the reason conjunction is also called a logical AND.

Truth table

Let $p$ and $q$ be arbitrary propositions. The exact definition of conjunction is given by defining the truth value of $p \land q$ in all possible cases. This is done by specifying a truth table. Let’s use T to represent true and F to represent false. A truth table enumerates all possible cases of the truth values of $p$ and $q$ and describes what the truth value will be for $p \land q$.

The truth table defining conjunction is given below:

$p$	$q$	$p \land q$
T	T	T
T	F	F
F	T	F
F	F	F

In computer science, the truth values are often represented by $0$ and $1$. Where $0$ represents false, and $1$ represents true. We call this the $0/1$-notation. In this notation, $p \land q$ is defined by the following table:

$p$	$q$	$p \land q$
$1$	$1$	$1$
$1$	$0$	$0$
$0$	$1$	$0$
$0$	$0$	$0$

A keen observer must have realized that each entry in the last column can be obtained by multiplying the corresponding entries in the first two columns. For this reason $p \land q$ is sometimes denoted by $pq$, where the symbol of multiplication is normally skipped.

Circuit diagram

Now, let’s illustrate the concept of conjunction through circuit diagrams. Here is the key that will help you in understanding them:

When both $p$ and $q$ are false, both switches are off, and no electricity reaches the bulb in the following circuit diagram. Therefore, the bulb is off, indicating that $p \wedge q$ is false.

When $p$ is true and $q$ is false, the circuit is not complete and the bulb is off. This indicates that $p \land q$ is false.

Similarly, when $p$ is false and $q$ is true, the circuit is not complete and the bulb is off. This indicates that $p \land q$ is false.

Finally, when both $p$ and $q$ are true, both switches are on, and the circuit is complete. The bulb is on, indicating that $p \land q$ is true.