The study of logic statements, holding values true or false, is called Boolean algebra. There are two types of connective logic that are important to learn. The first is conjunction. A conjunction statement is a statement involving an and. For example, a square has four equal sides and four equal measure angles. If a four-sided polygon does not have four equal sides and four equal measure angles, then it is not a square. Both sides of the and expression must have either the value of being true or false. To write an and statement using mathematical notation, use the {eq}\wedge {/eq} symbol. If p and q are statements with value either true or false, then the conjunction of p with q is written {eq}p \wedge q {/eq}. It is important to note that the conjunction of two statements is another statement, which also has the value: true or false. Consider another example. The conjunction of the statement "The sky is blue" with "the grass is green" is "The sky is blue, and the grass is green."
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The conjunction statement of two statements p and q is {eq}p\wedge q {/eq}, as briefed earlier. This means "p and q." The conjunction statement itself is either true or false. It is only true when both p and q are true. If one of either p or q is false, then the statement is false. It makes sense that conjunction is known as an 'and' statement because p and q must be true for the statement to be true. For example, suppose that there is a four-sided polygon S that is a square. Then it is true that the S has four equal sides, and it is true S has four equal angles. Hence, it is true that S has four equal sides length AND four equal measure angles.
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If either a four-sided polygon has only four equal angles or four equal length sides, it is possible that they are not squares. It is only definitively true that the polygon is a square if it has four equal sides and four equal measure angles.
The other important form of connective logic is disjunction. A conjunction is a statement involving an or. For two statements p and q, it is written in mathematical notation as {eq}p \vee q {/eq}. Again, both p and q have to be either true or false statements. Again, the disjunction of two statements is another statement. For example, suppose that {eq}x^2 -4x + 3 = 0 {/eq}. Then, either x = 3 or x = 1. Consider another example of a disjunction. The disjunction of the statements "The stop light is green" and "the stop light is red" is "The stop light is red or the stop light is green."
A disjunction statement of two statements p and q is written as {eq}p \vee q {/eq}. This means p or q. Again, p and q both need to be statements with a true or false value. To evaluate a disjunction statement if it is true, either one or both of the statements ought to be true. In other words, either p or q is true. Consider the example again, supposing that {eq}x^2 - 4x + 3 = 0 {/eq}. It is true that either x = 3 or x = 1. To check this, {eq}3^2 -4*3 + 3 = 9 - 12 + 3 = -3 + 3 = 0 {/eq}, and {eq}1^2 - 4*1 +3 = 1 - 4 + 3= -3 + 3 = 0 {/eq}. Thus, statement x = 3 or x = 1 is true, because both x = 3 and x = 1 satisfy the quadratic equation.
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Remember, only one of the statements needs to be true for the disjunction statement to be true.
How do conjunction and disjunction statements differ? Consider the previous example with the statements "The stoplight is red" and "the stop light is green." The conjunction of the two statements is "The stop light is red and the stop light is green." The disjunction of the two statements is "The stop light is red, or the stop light is green." Each of these statements evaluates differently. For a conjunction statement to be true, both of the statements in the conjunction must be true. Thus, "the stop light is red and the stop light is green" is false. As stop lights work, they can only be red or green. Thus, it cannot be true that both the stoplight is red and green. Assuming the stoplight is not yellow, then the statement "the stop light is red or the stop light is green" is true. This is because it is true that either the stoplight is red or green.
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This example highlights the difference between conjunction and disjunction. In conjunction, both statements need to be true in order for the conjunction to be true. In a disjunction, only one of the statements needs to be true for the disjunction to be true.
Boolean algebra is the algebra of mathematical logic. In Boolean algebra, there are two main types of connective statements. These are conjunction and disjunction. A conjunction statement is an 'and' statement. For a conjunction statement to be true, both of the statements must be true. For statements p and q, the conjunction is written using mathematical notation as {eq}p \wedge q {/eq}. This means p and q. The conjunction of two statements such as "pie is tasty" and "cake is yummy" is the statement "pie is tasty, and cake is yummy."
A disjunction statement is an OR statement. For a disjunction statement to be true, only one of the statements needs to be true, or both statements can be true. For statements p and q, the disjunction is written using mathematical notation as {eq}p \vee q {/eq}. This means p or q. The disjunction of two statements such as "the bowl is empty" and "the bowl is filled" is the statement "the bowl is empty, or the bowl is filled."
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A conjunction statement is a statement using 'and'. The conjunction is only valid when both of the statements in the conjunction are true.
A disjunction statement is a statement using 'or'. The disjunction statement is true if one or both parts of the disjunction are true.
Suppose there is a polygon, which is a square. Then, it is true that "the polygon has four equal length sides and four equal angles."
A simple example of a disjunction is the statement x < 0 or x > 1. It means x is less than 0 or x is greater than 1.
To create a conjunction statement, join two statements using 'and'. Consider these two statements: "the cow goes moo" and "the dog goes bark." The conjunction of these two statements is "the cow goes moo, and the dog goes bark."
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