Connecting two statements with "and" means both statements must be valid for the whole compound statement to be true.
Specific rules concerning the application of connective "AND".
When all the component statements joined by 'and, are correct, then the given statement is also true.
Any component statements connected by the connective 'and' is false; then, the entire compound statement is false.
Consider the following statement:
R: A rectangle has four sides, and the lengths of its opposite sides are equal.
This statement takes the connective 'and' to join two different mathematically acceptable observations. If we break this statement into component statements we have:
x: A rectangle has four sides
y: The lengths of its opposite sides are equal.
Here, both the component statements are true mathematically; therefore, statement R is also true.
Mentioned below are few examples to learn it more beneficially:
If the connector linking two statements is "or," it is a disjunction. In the aforementioned case, only one statement in the compound statement needs to be valid for the entire compound statement to be true.
Specific rules regarding the use of connective 'OR'
If any of the compound statements connected through Or is true, then the given compound statement is also true.
If all of the compound statements are connected through and is false, then the entire statement is false.
Let's have a quick look at the following statements:
R: The sum of two integers can be positive or negative.
The component can be read as:
a: The addition of two integers can be positive.
b: The addition of two integers can be negative.
In the statement, 'Or' is used as a connective, if each of the statements is true, then P is true. Here both 'a' and 'b' are true; consequently, P is true.
The connective 'Or' can either be inclusive or exclusive.
S: The length and breadth of a rectangle are 3 and 6, respectively.
This statement P can be broken as:
a: The length of a rectangle is 3.
b: The breadth of a rectangle is 6.
These component statements, i.e., a and b putting together, give us the statement S.
However, it can be observed that the component ‘b’ is not correct; therefore, as the given statement takes the connective and, the provided compound statement is false.
P: You can go in the northeast direction or southwest direction.
The statement implies you can only choose one direction, either northeast or southwest, but not both.
This statement uses the exclusive 'Or'.
Q: The applicants who have obtained 75% or eight pointers are eligible for the project.
This statement uses inclusive 'Or'.
A statement is said to be simple if further it cannot be broken down into more straightforward statements, that is if it is not composed of two or more than two simpler statements, joined by connectives.
1: What is a Statement in Mathematical Reasoning? Also State What is a Statement in a Proof?
Sol: A statement is called a mathematically acceptable statement if it is either true or false, but not both. Also, each of these statements is termed to be a compound statement. Furthermore, the compound statements are joined by the word “and” (^) the resulting statement is called conjunction denoted as - a ^ b.
A logical argument that confirms a specific statement, proposition, or mathematical formula is correct is referred to as proof. Besides, it contains a set of presumptions termed as axioms, connected by statements of deductive reasoning termed as an argument to drive the proposition that is being proved.
2: Analyze the statement, given that people who require home and consolation are apt to do odd things; it is clear that people who are apt to do odd things require home and consolation. Is his statement, of the form (P⇒Q)⇒(Q⇒P), logically equivalent to people?
A: Those who require home and consolidation are not apt to do odd things.
B: Those who require home and consolidation are apt to do odd things.
C: Those who are apt to do odd things require home and consolidation.
D: Those that are apt to do odd things if and only if they require home and consolidation.
Sol: Option C.
People who can do odd things require home and consolidation. The given statement is “people who require home and consolation are apt to do odd things”.
It is in the form of p⇒q, where p is “in need of home and consolation,” and q is “apt to do odd things”.
So q⇒p is equal to “people who are apt to do odd things need home and consolation”.
Therefore ‘option C’ is the correct answer.
We hope, with this lesson, that you have got a clear vision of the concept of compound and simple statements, connectives and its types, and concepts related to the same topic.