![]() In the truth table above, when p and q have the same truth values, the compound statement (p q) (q p) is true. Let's look at a truth table for this compound statement. In the first conditional, p is the hypothesis and q is the conclusion in the second conditional, q is the hypothesis and p is the conclusion. The compound statement (p q) (q p) is a conjunction of two conditional statements. Given:ĭetermine the truth values of this statement: (p q) (q p) ![]() Biconditional Statement Problems With Interactive ExercisesĮxample 1: Examine the sentences below. To prove this, we will use some of the above-described laws and from this law we have: Hence we can say that P ↔ Q ? (P → Q) ∧ (Q → P).Įxample 3: In this example, we will use the equivalent property to prove the following statement: This table contains the same truth values in the columns of P ↔ Q and (P → Q) ∧ (Q → P). Hence we can say that p → q ? ¬p ∨ q.Įxample 2: In this example, we will establish the equivalence property for a statement, which is described as follows: This table contains the same truth values in the columns of p → q and ¬p ∨ q. We will prove this with the help of a truth table, which is described as follows: P Some of them are described as follows:Įxample 1: In this example, we will establish the equivalence property for a statement, which is described as follows: There are various examples of logical equivalence. Similarly, this table also contains the same truth values in the columns of P ∧ (P ∨ Q) and P. This table contains the same truth values in the columns of P ∨ (P ∧ Q) and P. The following notation is used to indicate the identity law: Suppose there is a compound statement P, a true value T and a false value F. That means if we combine a statement and a True value with the symbol ∧(and), then it will generate the statement itself, and if we combine a statement and a False value with the symbol ∨(or), then it will generate the False value. ![]() Similarly, we will do this with the opposite symbols. If we combine a statement and a False value with the symbol ∧(and), then it will generate the statement itself. According to this law, if we combine a statement and a True value with the symbol ∨(or), then it will generate the True value. Same as we can prove P ∧ (Q ∨ R) ? (P ∧ Q) ∨ (P ∧ R)Ī single statement is used to show the identity law. Hence we can say that P ∨ (Q ∧ R) = (P ∨ Q) ∧ (P ∨ R) This table contains the same truth values in the columns of P ∨ (Q ∧ R) and (P ∨ Q) ∧ (P ∨ R). The following notation is used to indicate the idempotent law: According to this law, if we combine two same statements with the symbol ∧(and) and ∨(or), then the resultant statement will be the statement itself. In the idempotent law, we only use a single statement. There are various laws of logical equivalence, which are described as follows: ![]() Here, AND is indicated with the help of ∧ symbol and OR is indicated with the help of ∨ symbol. In this law, we will use the 'AND' and 'OR' symbols to explain the law of logical equivalence. With the help of the logical equivalence definition, we have cleared that if the compound statements X and Y are logical equivalence, in this case, the X ⇔ Y must be Tautology. So X = Y or X ⇔ Y will be the logical equivalence of these statements. With the help of symbol = or ⇔, we can represent the logical equivalence. Suppose there are two compound statements, X and Y, which will be known as logical equivalence if and only if the truth table of both of them contains the same truth values in their columns. Next → ← prev Law of Logical Equivalence in Discrete Mathematics ![]()
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