Profile
Settings
Refer your friends
Sign out
Introduction
A mathematical statement is the fundamental unit of mathematical reasoning. Let’s start with a few sentences:
- In 2003, India’s president was a woman.
- An elephant is far larger than a human.
After reading these sentences, we immediately determine that the first sentence is untrue and the second is correct. There is no ambiguity in this regard. Statements are the term used in mathematics to describe such utterances.
Consider the following sentence, for example:
- Women have a higher IQ than men
Some may believe it to be real, while others may disagree. We can’t say if this sentence is always true or false in this case. This indicates that the sentence is ambiguous. As a mathematical statement, such a sentence is unacceptable.
If a sentence is either true or untrue but not both, it is called a mathematically acceptable statement. Every statement we make here is one that is “mathematically acceptable.”
Compound Statements
Many mathematical statements can be made by connecting one or more statements using words like “and,” “or,” and so on. Take a look at the following statement p:
Something is amiss with the light bulb or the wiring.
This comment implies that there is a problem with the bulb or with the wiring. That is to say; the following statement is made up of two smaller statements:
q: There’s a problem with the light bulb.
r: There is a problem with the wiring.
“Or” connects them.
Now consider the following two statements:
p: The number seven is an odd number.
q: The number seven is a prime number.
With the word “and,” these two sentences can be joined.
r: The number 7 is both an odd number and a prime number.
This is an example of a compound statement.
Compound statements definition
Based on the above two examples, we can define compound statement as follows :
“A compound statement is one that is composed of two or more statements. Each statement in this scenario is referred to as a compound statement.”
Compound Statement Rules
- Rules for Connective “And” in a Compound Statement
There are several rules to follow when using connective “And” in a compound statement:
- I) The provided statement is true if all of the component statements connected by ‘and’ are true.
(ii) The entire compound statement is false if any of the component statements connected by the connective ‘and’ is false.
Consider the following proposition:
P: A square has four sides, and they all have the same length.
This statement takes the connective and combines two different mathematically acceptable statements. If we break this statement down into its constituent statements, we get:
a: There are four sides to a square.
b: A square’s sides are all the same length.
Because both of the component statements are mathematically correct, statement P is correct as well.
Example:
Let’s look at another example to help you grasp it better:
P: The squares of 3, 4, 5, and 6 are correspondingly 8, 16, 25, and 36.
This statement P can be divided into four parts:
a: The square of three is eight.
b: The square of four is sixteen.
c: The square of five is twenty-five.
d: 6 squared equals 36.
When these component statements are combined, we get the statement P.
However, component a is incorrect; hence the given compound statement is untrue because the supplied statement takes the connective ‘AND’
-
Rules for Connective “Or” in a Compound Statement
There are several rules to follow when using connective “Or” in a compound statement:
- I) The provided compound statement is true if any of the component statements connected by Or is true.
(ii) The complete statement is false if all of the component statements are connected by the connective and are false.
Take a look at the following assertions:
P: A positive or negative sum of two integers is possible.
The component can be specified as follows:
a: A positive sum of two integers is possible.
b: The difference between two integers can be negative.
Because Or is used as a connective in the sentence, P is true if either of the statements is true. P is true because both a and b are true.
The ‘Or’ statement can be inclusive or exclusive.
Example:
P: You have the option of heading east or west.
This means that you can only go one way, either east or west, and not both. This is an exclusive statement.
Or
S: Candidates with a score of 75% or 8 points are eligible for the position.
This sentence is either inclusive or exclusive.
Practice Compound Statement Examples
Now that we have learnt the rules, here are a few compound statement examples for practice:
- I) A square is a quadrilateral with four sides that are of the same length.
(ii) An MCA can be pursued by someone who has taken Mathematics or Computer Science.
(iii) Haryana and Uttar Pradesh share a capital, Chandigarh.
(iv) 2 is either a rational or irrational number.
(v) Twenty-four is a multiple of two, four, and eight.
Solution for compound statement examples
I )The component statements are:
p: A square is a quadrilateral with four sides and
q: All four sides of the square are of the same length.
Both of these assertions are accurate, as we know. The connecting word is ‘and’ in this case.
Hence the compound statement is correct.
- ii) The component statements are as follows:
p: A person who has taken Mathematics can pursue an MCA degree.
q: A person who has taken computer science can pursue an MCA degree.
Both of these assertions are correct. The connecting word is ‘or’ in this case. Hence the compound statement is correct
(iii) The component statements are as follows:
p: Chandigarh is Haryana’s capital.
q: Chandigarh is the state capital of Uttar Pradesh.
The first statement is correct; nevertheless, the second is incorrect. The connecting word is ‘and’ in this case, therefore the compound statement is incorrect.
- iv) Consider the next statements:
p: The number 2 is rational.
q: The number 2 is irrational.
The first statement is correct, whereas the second is incorrect. The connecting word is ‘or’ in this case. Therefore the compound statement is correct
(vi) The statements that make up the component statements are as follows:
p: 24 is a multiple of 2.
q: 24 is a multiple of 4.
r: A multiple of eight is 24.
Each of the three assertions is correct. The connecting words are ‘and’ in this case. Hence the compound statement is correct.
Conclusion
Compound statements are an important topic and make up a majority portion of Mathematical Reasoning. It is made up of two or more statements, and each statement is known as a constituent statement. These statements are generally connected by the words “and” or “or”. Both these connectivity follows some special rules which need to be learnt and applied while solving questions.
