Comparing Conjunctions and Disjunctions in a Truth Table

By Mary Jane Sterling

When studying logic in your finite mathematics course, you will probably work with truth tables. A truth table is a visual representation of all the possible combinations of truth values for a given compound statement.

Two types of connectives that you often see in a compound statement are conjunctions and disjunctions, represented by ∧ and ∨, respectively.

It’s important to know the difference between these two connectives.

Identifying a conjunction

The conjunction, p q, puts the word and between two statements to create a compound statement.

Consider the following statements:

(1) Chicago is a city in Illinois.

(2) Red is a color in the American flag.

(3) 7 + 3 = 11.

(4) San Francisco is a city in Florida.

Statements (1) and (2) are true, and Statements (3) and (4) are false.

Next, you construct a truth table for the conjunction p ^ q.

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Referring to the first line of Ts and Fs in the table, when both statements are true, their conjunction p ^ q is true. For example, using Statements (1) and (2), the conjunction reads: “Chicago is a city in Illinois, and red is a color in the American flag.”

The second line of Ts and Fs says that when the first statement is true and the second is false, their conjunction p ^ q is false. Using Statements (1) and (3), the conjunction reads: “Chicago is a city in Illinois, and .”

In the third line, when the first statement is false and the second statement is true, their conjunction p ^ q is false. Using Statements (4) and (2), the conjunction reads: “San Francisco is a city in Florida, and red is a color in the American flag.”

And, finally, when both statements are false, their conjunction is false. Using Statements (3) and (4), the conjunction reads: “7 + 3 = 11, and San Francisco is a city in Florida.”

Basically, what you see here is that for a conjunction to be true, both of the component statements have to be true.

Identifying a disjunction

The disjunction, p q, uses the word or to create a compound statement.

The truth table for the disjunction is shown here:

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For a disjunction to be true, only one of the component statements needs to be true. Consider the following compound statements representing the four rows.

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Both are true; the compound statement is true.

TF: “It rains in Hawaii, or all cows have seven legs.” The first statement is true, so the compound statement is true.

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The second statement is true, so the compound statement is true.

FF: “All cows have seven legs, or pigs can fly.” Both statements are false, so the compound statement is false.