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High School Mathematics - 2
14.1 Introduction to Mathematical Logic

The word "logic" is derived from the Greek word logos, which means reason or discourse. This reasoning is usually expressed in terms of declarative sentences. These are referred to as statements.

Statements
These are assertions in words or symbols which are either true or false but not both simultaneously.
Simple and Compound statements
A statement is said to be simple if it cannot be broken down to two or more sentences while a sentence that can be broken into sub statements is called a compond sentence.
Truth and Falsity of statements
The truth value of statements (true or false) is referred to as truth and falsity of statements.
Basic Logical connectives
There are many ways in which simple sentences can be combined to form compound statements. Such combinations are formed by using words called connectives to join the statements. Two of the most important connectives are the words and and or. In the study of logic, the word and is symbolised by '^' and the word or by' v'.
Conjunction: If two statements are combined by the word "and" the statement is called conjunction.
Disjunction: If two statements are combined by the word "or" the result is a disjunction.
Negation: The negation of a conjunction p^q is the disjunction of the negation of p and the negation of p and the negation of q, that is, ~(p^q) = ~p v ~q.
Example: Let p:3+4 and q: 6< 9
~(p^q) = 3+47 or 6< 9
Negation of disjunction: A disjunction p v q means that either p or q or both exist. Therefore, the negation of disjunction would mean the negation of both p and q simultaneously. Therefore
~(pvq) = ~p ^~q
Example: p: 5 is greater than 3, q: 4 is less than 8
Then ~(pvq) = 5 is not greater than 3 and 4 is not less than 8.
Negation of a negation: Negation of a negation is the statement itself.
Example: p: Crows are black.
~ (~p) = Crows are black.


Directions: Answer the following.
Q 1: Find out if the sentence is a statement or not. The number 17 is prime.
Yes
No

Q 2: Find out if the sentence is a statement or not. All integers are natural numbers.
Yes
No

Q 3: Find out if the sentence is a statement or not. The number 6 has 3 prime factors.
No
Yes

Q 4: Find out if the sentence is a statement or not. Listen to me, John!
No
Yes

Q 5: Find out if the sentence is a statement or not. 2+1 = 3
Yes
No

Q 6: Find out if the sentence is a statement or not. 2+3 < 6
Yes
No

Q 7: Find out if the sentence is a statement or not. Zero is a complex number.
No
Yes

Q 8: Find out if the sentence is a statement or not. Violets are blue.
No
Yes

Question 9: This question is available to subscribers only!

Question 10: This question is available to subscribers only!


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