## Problems based on Converse, inverse and ContrapositiveIf we want to learn the converse, inverse, and contrapositive statements, we have to see our previous article, Logical Connectives. ## Logical ConnectivesLogical connectives are a type of operator which is used to combine one or more than one propositions. There are basically 5 types of connectives in propositional logic. In this section, we are going to learn about the converse, inverse, and contrapositive of conditional statements. ## Converse, Inverse and ContrapositiveIf there is a conditional statement x → y, then - The converse statement will be y → x
- The inverse statement will be ∼x → ∼y
- The contrapositive statement will be ∼y → ∼x
## Important Notes:There are some important points that we should keep in our mind, which are described as follows: ## Note 1: We can only write the converse, inverse, and contrapositive statements only for the conditional statements x → y.## Note 2: If we perform two actions, then the output will always be the third one.
- Contrapositive can be described as a inverse of converse.
- Converse can be described as an inverse of contrapositive.
- Contrapositive can be described as a converse of inverse.
- Inverse can be described as a converse of contrapositive.
- Converse can be described as a contrapositive of inverse.
- Inverse can be described as a contrapositive of converse.
For a conditional statement x → y, There will be an equal result between its converse statement (y → x) and the inverse statement (∼x → ∼y). There will also be the same result between x → y and its contrapositive statement (∼y → ∼x). ## Problem-based on Converse, Inverse and ContrapositiveThere are some problems on the basis of the converse, inverse, and contrapositive, and we will show some of them like this:
Here we will write the converse, inverse, and contrapositive of some statements, which are shown below: - If the weather is sunny, then I will go to school.
- If 3y - 2 = 10, then x = 1.
- If there is rainy weather, then I will go outside to enjoy it.
- You will get good marks only if you study hard.
- I will go to the market if my cousins come.
- I go to college whenever my friends come.
- I will give you a party only if I buy a good dress.
- If I become famous, then I will earn a lot of money.
We have the following details: The given statement is, "If the weather is sunny, then I will go to school." This statement must have the form: "if x then y". So, this statement contains a symbolic form, i.e., x → y, where x: The weather is sunny y: I will go to school
We have the following details: The given statement is, "If 3a - 2 = 10, then a = 1." This statement must have the form: "if x then y". So, this statement contains a symbolic form, i.e., x → y, where x: 3a - 2 = 10 y: a = 1
We have the following details: The given statement is, "If there is rainy weather, then I will go outside to enjoy it." This statement must have the form: "if x then y". So, this statement contains a symbolic form, i.e., x → y, where X: There is rainy weather Y: I will go outside to enjoy it
We have the following details: The given statement is, "You will get good marks only if you study hard." This statement must have the form: "x only if y". So, this statement contains a symbolic form, i.e., x → y, where X: You will get good marks Y: You study hard
We have the following details: The given statement is, "I will go to the market if my cousins come." This statement must have the form: "y if x". So, this statement contains a symbolic form, i.e., x → y, where X: My cousins come Y: I will go to the market
We have the following details: The given statement is, "I go to college whenever my friends come." In this statement, "whenever" can be replaced with "if". After replacing the sentence will be - "I go to college if my friends come" So, this statement contains a symbolic form, i.e., x → y, where X: My friends come Y: I go to college
We have the following details: The given statement is, "I will give you a party only if I buy a good dress." This statement must have the form: "x only if y". So, this statement contains a symbolic form, i.e., x → y, where X: I will give you a party only Y: I buy a good dress
We have the following details: The given statement is, "If I become famous, then I will earn a lot of money." This statement must have the form: "If x then y". So, this statement contains a symbolic form, i.e., x → y, where X: I become famous Y: I will earn a lot of money
Here we have to determine the converse a statement, i.e., "I go to school only if the weather is sunny" among all the given statements. - I go to school if the weather is sunny
- If I go to school, then the weather is sunny
- If the weather is not sunny, then I do not go to school.
- If I do not go to school, then the weather is sunny.
We have the following details: The given statement is, "I go to school only if the weather is sunny." This statement must have the form: "x only if y". We can also write it as "If x then y". So, this statement contains a symbolic form, i.e., x → y. The converse of this form will be y → x, where X: I go to school Y: The weather is sunny As we know that the converse statement of the given statement will be "If the weather is sunny, then I go to school", which is in the form "if y then x". - The
**first statement**is**true**. The first statement is, "I go to school if the weather is sunny". This statement is in the form "x if y". We can also write it as "if x then y", which indicates that "If the weather is sunny, then I go to school", which is the converse of a given statement. That's why the first statement is true. - The
**second statement**is**false**. The second statement is, "If I go to school, then the weather is sunny" and this statement is in the form "if x then y". The second statement is already given in the question. That's why it is not true. - The
**third statement**is**false**. The third statement is, "If the weather is not sunny, then I do not go to school". This statement is in the form "∼y → ∼x". It is not the converse because this statement is the inverse of statement given in the question. That's why this statement is not true. - The
**fourth statement**is**false**. The fourth statement is, "If I do not go to school, then the weather is sunny". This statement is in the form "∼x → y. This form is something different because it is neither inverse nor converse nor contrapositive. This is because one side is negative, and the other side is not negative, so it will not fit in any of the categories. That's why this statement is not true.
Hence, option (A) is true. |