Disjunction

What is a disjunction?

Disjunction is also a binary operation (requiring two operands). When two propositions are connected using the disjunction operation, it becomes a new proposition whose truth value depends on the truth values of the operands. Disjunction is also known as the OR operator.

Example

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Consider the following propositions:

• $S_A$: Sara is taking an algebra course.
• $S_C$: Sara is taking a calculus course.

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Let’s make a new proposition $S$ by taking the disjunction of $S_A$ and $S_c$.

• $S$: “Sara is taking an algebra course” or “Sara is taking a calculus course.”

We can simplify $S$ without changing the meaning.

• $S$: Sara is taking an algebra or a calculus course.

We use the $\lor$ symbol to represent the disjunction operation. Therefore,

$$S \equiv S_A \lor S_C.$$

The truth value of $S$ is false only when $S_A$ and $S_C$ both are false and it is true otherwise.

Truth table

We can represent this relationship of truth values in the form of a truth table. Let $p$ and $q$ be two propositions. The definition of their disjunction, $p \lor q,$ is given by the following truth table.

$p$	$q$	$p \lor q$
T	T	T
T	F	T
F	T	T
F	F	F

Note: $p \lor q$ is true when at least one of the $p$ or $q$ is true.

In $0/1$-notation, $p \lor q$ is defined by the following table:

$p$	$q$	$p \lor q$
$1$	$1$	$1$
$1$	$0$	$1$
$0$	$1$	$1$
$0$	$0$	$0$

A keen observer must have realized that each entry in the last column can be obtained by taking the maximum of corresponding entries in the first two columns; that is,

$$p\lor q = \mathrm{max}(p,q).$$

Circuit diagram

Let’s illustrate the concept of disjunction through circuit diagrams. Here is the key that will aid you in understanding diagrams.

When both $p$ and $q$ are false both switches are off. Therefore, the circuit is not complete, and the bulb is off. This indicates that $p \lor q$ is false.

When $p$ is true and $q$ is false, one of the switch is on and the other switch is off. Since the bulb is connected to both $+$ and $-$ terminals of the battery, therefore it is on. This indicates that the truth value of $p \lor q$ is true.

Similarly, when $p$ is false and $q$ is true, one of the switches is on and the other switch is off. Since the bulb is connected to both $+$ and $-$ terminals of the battery, therefore it is on. This indicates that the truth value of $p \lor q$ is true.

When both $p$ and $q$ are true, both switches are closed. Since the bulb is connected to both $+$ and $-$ terminals of the battery, therefore it is on. This indicates that the truth value of $p \lor q$ is true.

Inclusive versus exclusive disjunction

In mathematics, the convention is to always take disjunction operation as an inclusive disjunction. This is in contrast to the everyday use of the word “or.” In everyday language, “or” has two meanings: an inclusive one and an exclusive one. The exact meaning is extracted from the context.

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Consider these propositions:

• $k_1$: Kennedy is wearing sneakers.
• $k_2$: Kennedy is wearing slippers.
• $k \equiv k_1 \lor k_2$: Kennedy is wearing sneakers or slippers.

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When we use $k$ as a statement in everyday language, we never assume that Kennedy could be wearing both sneakers and slippers. The logical OR or disjunction does not prohibit the possibility that $k_1$ and $k_2$ are simultaneously true. In this case, the disjunction does not correspond to our intended meaning. We want to say that either $k_1$ is true or $k_2$ is true, but not both; we are demanding an exclusive disjunction.

In everyday language, the meaning of the word “or” is extracted from the context.

Mathematicians are meticulous and do not want to leave any ambiguity in expressing mathematical statements. Therefore, they have agreed that $\lor$ would always represent an inclusive disjunction. This definition makes children very happy: for Timmy, when asked to eat a chocolate ice cream or a strawberry one, he can eat both!

When reading mathematical text always keep in mind that $\vee$ is an inclusive disjunction.

In mathematics, we use the symbol $\oplus$ to represent exclusive disjunction. It is called exclusive OR, and we also write it as XOR.

Let’s use exclusive disjunction to construct a new proposition $k'$.

$$k' \equiv k_1 \oplus k_2.$$

$k'$ now properly expresses what we want. Translated in common language $k'$ states that,

“Kennedy is wearing sneakers or slippers but not both.”

Given two propositions $p$ and $q$, their exclusive disjunction, $p \oplus q$, is defined by the truth table below. We have also included the definition of the inclusive disjunction for comparison.

$p$	$q$	$p \lor q$	$p \oplus q$
T	T	T	F
T	F	T	T
F	T	T	T
F	F	F	F

Note: When both $p$ and $q$ are true then their XOR is false. This is sometimes the case in real life. In such cases, “or” is exclusive, and therefore it is actually an XOR.

We must be careful, in mathematics and everyday life while using disjunction and be clear if we mean inclusive or exclusive disjunction.