In mathematical logic, words have precise meanings. Logic attempts to show truthful conclusions emerging from truthful premises, or it identifies falsehoods reliably. Mathematical and logical statements are joined with connectors; conjunctions and disjunctions are two types of logical connectors.
With logic, statements can be labeled as true or false, such as:
Clearly, some of those six statements are false, but the point is, they are testable claims. They do not express opinion. Contrast them with, say, "I like cheeseburgers," which shows an opinion.
Statements are often symbolized with the letters p and q. They are strung together using connectors, so you can combine ideas using "and," or "or" between statements. Two statements joined with connectors create a compound statement. Joining logical statements is not the same as stringing together ideas in ordinary English conversation. Compare:
The first connected statements, a single compound statement, are opinions. The second compound statement is a logical statement (but the compound statement is false).
The two types of connectors are called conjunctions ("and") and disjunctions ("or"). Conjunctions use the mathematical symbol and disjunctions use the mathematical symbol .
Joining two statements with "and" is a conjunction, which means both statements must be true for the whole compound statement to be true. Conjunctions are symbolized with the character, so these two discrete statements can be combined in a compound statement:
Only if both parts of the compound statement are true is the entire statement true.
When the connector between two statements is "or," you have a disjunction. In this case, only one statement in the compound statement needs to be true for the entire compound statement to be true.
Let's look at our original statements again:
If we link one true and one false statement into a compound statement using the connector "or," (symbolized by ) we still have a true compound statement:
Here are four other compound statements taken from our original statements. Determine the symbols and if the compound statements are true or false:
Did you say pq, and did you rate this as true? Both statements are true, so the compound statement joined by "and" is true.
Did you say pq, and rate this compound statement as true? Though quadrilaterals do not have 11 sides, the conjunction "or" makes the compound statement true since some quadrilaterals are parallelograms.
Did you say pq? More important, did you say this compound statement was false? Since quadrilaterals do not have 11 sides, the conjunction is false.
Did you write pq? Did you say this is false, since both sides of the compound statement are false?
Conjunctions and disjunctions are ways of joining logical statements, with every joined, compound statement either true or false. For conjunctions, both statements must be true for the compound statement to be true. For disjunctions, only one statement needs to be true for the compound statement to be true.
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