## Equivalence of Formula in Discrete mathematicsSuppose there are two formulas, X and Y. These formulas will be known as equivalence iff X ↔ Y is a tautology. If two formulas X ↔ Y is a tautology, then we can also write it as X ⇔ Y, and we can read this relation as X is equivalence to Y. ## Note: There are some points which we should keep in mind while linear equivalence of formula, which are described as follows:- ⇔ is used to indicate only symbol, but it is not connective.
- The truth value of X and Y will always be equal if X ↔ Y is a tautology.
- The equivalence relation contains two properties, i.e., symmetric and transitive.
## Method 1: Truth table method:In this method, we will construct the truth tables of any two-statement formula and then check whether these statements are equivalent.
As we can see that X ∨ Y and ¬(¬X ∧ ¬Y) is a tautology. Hence X ∨ Y ⇔ ¬(¬X ∧ ¬Y).
As we can see that X → Y and (¬X ∨ Y) are a tautology. Hence (X → Y) ⇔ (¬X ∨ Y) ## Equivalence formula:There are various laws that are used to prove the equivalence formula, which is described as follows:
## Method 2: Replacement ProcessIn this method, we will assume a formula A : X → (Y → Z). The formula Y → Z can be known as the part of formula. If we replace this part of the formula, i.e., Y → Z, with the help of equivalence formula ¬Y ∨ Z in A, then we will get another formula, i.e., B : X → (¬Y ∨ Z). It is an easy process to verify whether the given formulas A and B are equivalent to each other or not. With the help of replacement process, we can get B from A.
Now we will use the Associative law like this: Now we will use De Morgan's law like this: Hence proved
Hence proved {(X → Y) ∧ (Z → Y)} ⇔ (X ∨ Z) → Y
Hence proved
Now we will use the Associative and Distributive laws like this: Now we will use De Morgan's law like this: Now we will use the Distributive law like this: Hence proved
First, we will use De Morgan's law and get the following: Therefore, Also Hence Thus Hence we can say that the given formula is a tautology.
Now we will use De Morgan's law like this: Now we will use the Associative law and Commutative law like this: Now we will use the Negation law like this: Hence we can say that the given formula is a tautology.
- Marry will complete her education or accept the joining letter of XYZ Company.
- Harry will go for a ride or run tomorrow.
- If I get good marks, my cousin will be jealous.
1. Suppose X: Marry will complete her education. Y: Accept the joining letter of XYZ Company. We can use the following symbolic form to express this statement: The negation of X ∨ Y is described as follows: In conclusion, the negation of given statement will be: 2. Suppose X: Harry will go for a ride Y: Harry will run tomorrow We can use the following symbolic form to express this statement: The negation of X ∨ Y is described as follows: In conclusion, the negation of given statement will be: 3. Suppose X: If I get good marks. Y: My cousin will be jealous. We can use the following symbolic form to express this statement: The negation of X → Y is described as follows: In conclusion, the negation of given statement will be:
- I need a diamond set and worth a gold ring.
- You get a good job or you will not get a good partner.
- I take a lot of work and I can't handle it.
- My dog goes on a trip or it makes a mess in the house.
- I don't need a diamond set or not worth a gold ring.
- You cannot get a good job and you will get a good partner.
- I do not take a lot of work or I can handle it.
- My dog not goes on a trip and it does not make a mess in the house.
- If it is raining, then the plan to go to the beach is cancelled.
- If I study hard, then I will get good marks on the exam.
- If I go to a late-night party, then I will get punishment by my father.
- If you don't want to talk to me, then you have to block my number.
- If the plan to go to the beach is cancelled, then it is raining.
- If I get good marks on the exam, then I study hard.
- If I will get punishment by my father, then I go to a late-night party.
- If you have to block my number, then you don't want to talk to me.
And
In this truth table, we can see that the columns of (X → Y) → Z and X → (Y → Z) do not contain identical values. |