Logic
The thought process that humans implement in reaching a conclusion is called logic.
In Mathematics, we require logic to validate given statements. For example, one can say, "The sum of two non-zero and positive integers is always an even integer." One cannot tell if this claim is either true or false without logic. The path we take to approve or disprove this statement is what we are going to study in this chapter.
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Grade 11 Basic Mathematics Logic Exercise Complete Solutions
First, we shall look at the basic terminologies used in logic:
Statements
A statement is any claim or hypothesis that can either be true or false, but not both. In simpler words, statements are neutral sentences that do not involve human-perceived emotions such as beauty, hate, etc. that differ from person to person.
Examples of statements:
- 2 + 2 = 4. It is a statement because it is either true or false. It cannot be that 2+2 = 4 and at the same time 2 + 2 is not equal to 4.
- Kathmandu is the capital of the United World. It is a statement because it is either true or false. Either Kathmandu is the capital city of the mentioned country or it isn't, but it cannot be both at the same time.
- Rose is beautiful. It involves human-perceived emotions. A rose may look beautiful to one but not to the other. It depends on human understanding and behaviors. So, it is not a statement.
Simple Statement
A simple statement makes a single claim only.
- 2 * 4 = 8. It is a simple statement because it presents only one claim.
Compound Statement
- 7 is greater than 6 and 7 is less than 8. It is a compound statement because we can break this statement into two simple statements: a) 7 is greater than 6, and b) 7 is less than 8.
Truth Value and Truth Table
A truth value is the result of a statement: either true or false. We can represent truth value in several ways: True or False, T or F, and 1 or 0.
An organization of all the truth values of a statement in the form of a table is called the truth table. This table gives us an idea of the conditions when a given statement is true and when it is false.
A truth table contains the statements and their corresponding truth values.
Logical Connectives
We can form a compound statement by joining two or more simple statements. The mathematical symbols used to join those simple statements are called logical connectives.
In this article, we will discuss the introduction, example, and truth table of some of the logical connectives:
Conjunction
Let us consider two simple statements p and q, then a compound statement is a new statement formed by joining the two statements with the word "and" and written as p^q.
The mathematical symbol of conjunction is ^.
Example: Let p = Kathmandu lies in Nepal and q = Kathmandu is the capital of Nepal. Then, their conjunction is Kathmandu lies in Nepal and Kathmandu is the capital of Nepal and is denoted by p^q.
Truth Table for Conjunction Statements
| p | q | p^q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
From the truth table, we can conclude that a compound statement formed by conjunction is true only when both statements are true.
Disjunction
Let us consider two simple statements p and q, then a compound statement is a new statement formed by joining the two statements with the word "or" and written as pVq.
The mathematical symbol of conjunction is V.
Example: Let p = Kathmandu does not lie in Nepal and q = Kathmandu is not the capital of Nepal. Then, their conjunction is Kathmandu does not lie in Nepal or Kathmandu is not the capital of Nepal and is denoted by pVq.
Truth Table for Disjunction Statements
| p | q | pVq |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
From the truth table, we can conclude that a compound statement formed by disjunction is false only when both statements are false.
Negation
Let us consider a simple statement p, then a new statement formed by the word "not" and written ~p is called the negation of the statement p.
The mathematical symbol of conjunction is ~.
Example: Let p = Kathmandu lies in Nepal. Then, its negation is Kathmandu does not lie in Nepal and is denoted by ~p.
Truth Table for Negation Statements
| p | ~p |
| T | F |
| F | T |
From the truth table, we can conclude that the truth value of a new statement formed by negation is always the opposite of the original statement.
Conditional (Implication)
Let us consider two simple statements p and q, then a compound statement is a new statement formed by joining the two statements with the phrase "if ... then" and written as p $\rm \implies $ q.
The mathematical symbol of conditional is $\rm \implies $.
Example: Let p = Kathmandu does not lie in Nepal and q = Kathmandu is not the capital of Nepal. Then, their conditional statement is If Kathmandu does not lie in Nepal then Kathmandu is not the capital of Nepal and is denoted by p $\rm \implies$q.
Truth Table for Disjunction Statements
| p | q | p$\rm \implies$q |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
From the truth table, we can conclude that a compound statement formed by disjunction is false only when the first statement is true and the second statement is false.
Biconditional (Equivalence)
Let us consider two simple statements p and q, then a compound statement is a new statement formed by joining the two statements with the phrase "if and only if" and written as p $\rm \iff $ q.
The mathematical symbol of biconditional is $\rm \iff $.
Example: Let p = Kathmandu lies in Nepal and q = Kathmandu is not the capital of Nepal. Then, their conditional statement is Kathmandu is the capital of Nepal if and only if Kathmandu lies in Nepal and is denoted by p $\rm \iff$q.
Truth Table for Biconditional Statements
| p | q | p$\rm \implies$q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
From the truth table, we can conclude that a compound statement formed by biconditional is true only if the truth values of both statements are the same. Otherwise, they are different.
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