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Compound Statements


Identifying Compound Statements
A compound statement is made up of two or more simple statements. Each statement forms a component of the compound statement. Generally “and” or “or” will be used in compound statements.

Let us see some examples.
  1. The sun rises in the east and the sky is blue.
  2. 9 is an odd number and 9 is a prime number.
  3. All integers are rationals or is an irrational number.
  4. All squares are rectangles or all squares are parallelograms.
We can break up these compound statements into their component statements, thus,


We notice that the connective words used are “and” and “or” is statements.

Now let us see how to find the truth value of a compound statement.

Example 1:
50 is divisible by 2, 5 and 10

Solution:
This is a compound statement. We can split it up as:
: 50 is divisible by 2
: 50 is divisible by 5
: 50 is divisible by 10
are true statements. So is true.

Example 2:
Consider the statement
All integers are rationals and all rationals are integers.

Solution:
Let us split the about statement into its components.
: All integers are rationals.
: All rationals are integers.
Here is true, is not true.
So we conclude that '' is not true.
  1. Rules for checking truth value of a compound statement connected by “and”.
    1. The compound statement with “and” is true if all the component statements are true.
    2. The compound statement with “and” is false, if any of the component statements is false.
That means, if there are two component statements, either of them is false or both of them are false will result in the compound statement becoming false.

Example 1: 
Write the component statements of the following compound statement and check whether the compound statement is true or false.
  1. All similar triangles are congruent and all congruent triangles are similar.
Solution:
Component statement are:
: All similar triangles are congruent
(This is false)
: All congruent triangles are similar
(This is true)
.
  1. 3 + 3 = 6 and is irrational.
Solution:
Component statements are
: 3 + 3 = 6 (True)
: is irrational (True)
Since is true and is true, '' is true.
  1. Every square matrix has an inverse and a transpose.
Solution:
Component Statements are:
: Every square matrix has an inverse (Not True)
: Every square matrix has a transpose (True)
is false; is true;  'and ' is false.
  1. Rules for checking the truth value of a compound statement connected by “or”.
    1. A compound statement with an “or” is true when one component
      statement is true or both the component statements are true.
    2. A compound statement with an “or” is false when both the component
      statements are false. (The rule is the same if there are more than two statements).
Note: The word “or” can be used in two different ways in statements.

Example 2:
Write the component statements of the following compound statement and check whether the compound statement is true or false.
  1. You have to produce your ration card or driving licience as proof of your residence.
Solution:
The compound statement when split into its components, we get
: you have to produce your ration card as proof of residence.
: you have to produce your driving licence as proof of residence.
A person may have both ration card and diving license. Here “or” is inclusive.
  1. Breakfast or lunch is free when you book a room in this hotel.
Solution:
Component Statements are:
: Breakfast is free when you book a room in this hotel.
: Lunch is free when you book a room in this hotel.
But both breakfast and lunch are not available free. Here "or" is exclusive.

Example 3:
Study the following compound statements identity the type of “or” used in each of them and also check the truth value of the statements.
  1. 1 is its own square or its own square not
  2. A rectangle is a quadrilateral or acquire.
  3. In Std XII, a student can take Biology or Computer science as an optional subject.
Solution:
  1. Components are:
    : one is its own square.
    : one is its own square not
    both and are true. Therefore, ' or ' is true
    Here “or” is inclusive
  2. The components are:
    : A rectangle is a quadrilateral
    : A rectangle is a square
    Here “or” is exclusive
    is true: is false. Therefore, 'or ' is true.
  3. Components are:
    : In std XII a student can take Biology as an optional.
    : In std XII a student can take computer science as an optional
    Hence “or” is exclusive.
    If and are true, ' or ' is true




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