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Compound Statements
In mathematics, a sentence that can be either true or false is a statement, and it cannot be true and untrue at the same time. Compound statements are groups of two or more statements that are connected using words like ‘or’, ‘and’, ‘if-then’, and ‘if and only if’. Now, let us learn more about compound statements.
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What Are Compound Statements?
Compound statements are a type of mathematical statement that is made up of two or more statements. Here, each of the sentences is known as an atomic statement. The words like ‘and’, ‘or’, ‘if them’, and ‘if and only if’ that link both statements, are linking words. These linking words are used to express the degree of association of the statements. For example, it is raining, and it is cold, where and is the linking word.
Let’s consider it is raining as p, and it is cold as q then the compound statement is represented as p v q, p ^ q, p ⇒ q, p ⇔ q. The symbols v, ^, ⇒, ⇔ represent the words ‘or’, ‘and’, ‘if-then’, and ‘if and only if’.
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What are the Types Of Compound Statements
Based on the connectives used, the compound statements can be classified into;
- Negation of a Statement
- Disjunction Statement
- Conjunction Statement
- Conditional Statement
- Bi-Conditional Statement
Negation of a Statement
The negation of a statement is the opposite or the negative of the statement. If the statement is P, then the negation is ~P. For example, P — it is raining then ~P — it is not raining.
Disjunction Statement
The disjunction statement is true if at least one of the statements is true. The connection word used here is ‘or’. The symbol used in the disjunction statement is “∨”. For example, it is raining, or it is sunny.
Conjunctions Statement
In conjunction statements, both statements are true and connected by AND. It is represented as P^Q. For example, she has a pen and a book.
Conditional Statement
In conditional statements, if the first statement is true, then the second statement must be true. Here the connection word is if then. The conditional statement is represented by ⇒. For example, if Tom studies well, then he will pass the test.
Bi Conditional Statement
Here the first statement is known as the antecedent and the second is known as the consequent. This means if both the statements are either true or both are either false. The connective used is if and only if, and represented as ⇔. For example, you are a teenager if and only if your age is between 13 and 19.
Truth Tables of Compound Statement
The truth table of compound statements is used to find the outcome of the compound statement based on their independent statements. The truth table is different for each type of compound statement.
Disjunction truth table: The connective used in the Disjunction statement is ‘or’, so the compound statement here is p v q. Based on the truth value of p and q.
| p | q | p v q |
| T | F | T |
| T | T | T |
| F | T | T |
| F | F | F |
Conjunction truth table: The connective used here is and to connect both statements, so the compound statement is p ^ q. If both individual statements are true then the compound statement is true. Based on the truth value of p and q.
| p | q | p ^ q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
Conditional truth table: In conditional statements, the connective used is if then, and represented by ⇒. The statement is false only if the hypothesis is true and the conclusion is flase. If both the hypothesis and conclusion are false, then the compound statement is true.
| p | q | p ⇒ q |
| T | F | F |
| T | T | T |
| F | T | T |
| F | F | T |
Biconditional truth table: The biconditional statement is connected using if and only if, and represented as ⇔.
| p | q | p ⇔ q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
Common Mistakes and How to Avoid Them in Compound Statement
Students make errors when working on compound statements. In this section, let’s learn a few common mistakes and ways to avoid them.
Mistake 1
Using double negation
Using double negation leads to confusion among students, such as “it is not true that I am not going” which creates confusion for the readers. So use a single negation to express the idea, as it gives clarity to the reader.
Mistake 2
Thinking both the statement must be true in disjunction mistakes
When working on disjunction statements where the connecting word is OR, students think that both statements should be true. But it is wrong the disjunction statement can be true even if one of the statements is true.
Mistake 3
Using conjunction in unrelated statements
Students sometimes join two statements with “AND” even when they are not logically connected, for example, “I will go to school and the sun is hot” which is meaningless. So when writing conjunction statements combining unrelated ideas can lead to confusing and meaningless statements. As each sentences gives the meaning to the overall statement.
Mistake 4
Thinking If means cause and effect in conditional statements
For sentences like, “If it rains, I will take an umbrella” which means the rain causes you to take the umbrella in a casual sense. But “if” cannot be the cause and effect all the time, this can lead to incorrect statements.
Mistake 5
Confusing with the types of conditional statements
Mixing up the simple conditionals and biconditionals can lead to errors. So it is important to understand the types of conditional statements as each of them is different.
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Solved Examples of Compound Statement
Problem 1
What is the compound statement that can be formed from the statements P: You go regularly to school and Q: You get good marks? Using: a) AND (∧) b) OR (∨) c) IF-THEN (⇒) d) IF AND ONLY IF (⇔)
Using and: You go regularly to school, and you get good marks
Using or: If you go regularly to school, or you get good marks
Using if-then: If you go regularly to school, then you get good marks
Using if and only if: You go regularly to school if and only if you get good marks
Explanation
In compound statements, we use the connectives like ‘and’, ‘or’, ‘if-then’, and ‘ if and only if’.
Problem 2
Construct the truth table for the statement: “If you study hard, then you will pass the exam.”
Here, P — you study hard
Q — you will pass the exam
| p | q | p ⇒ q |
| T | F | F |
| T | T | T |
| F | T | T |
| F | F | T |
Explanation
Here, the statement is in p ⇒ q format, so the only situation that makes the statement false is when p is true and q is face.
Problem 3
Identify whether the following statement is a conjunction, disjunction, conditional, or biconditional: “A triangle is equilateral if and only if all its sides are equal.”
The situation is biconditional
Explanation
The phrase if and only if is used as a connective, so it is a biconditional statement.
Problem 4
Are the statements “If I am hungry, then I eat” and “If I do not eat, then I am not hungry” logically equivalent? Justify your answer.
Yes, the statement is logically equivalent
Explanation
Here, P - I am hungry, and Q - I eat
The statements are
- If I am hungry, then I eat, and it is p ⇒ q
- If I do not eat, then I am not hungry, and it is ¬ Q ⇒ ¬ P
Problem 5
Write the negation of the statement: “If it is snowing, then the roads are slippery.”
It is snowing, and the roads are not slippery
Explanation
The statement given is that if it is snowing, then the roads are slippery
Here, P: It is snowing
Q: The roads are slippery
The negation of the condition is it is snowing and the roads are not slippery
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FAQs on Compound Statement
1.What is a compound statement?
2.What are the types of compound statements?
3.What is the difference between a conjunction and a disjunction?
4.How are the compound statements represented symbolically?
5.What are the examples of compound statements?
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