A Proposition is a declarative sentence that is either TRUE or FALSE, but never both.
Examples:
All the above are propositions as they can be true or false but not both.
Some examples of sentences that are not propositions:
All the above examples are not propositions, they can be either be true or false.
The negation operator is a unary operator which, when applied to a proposition p, changes the truth value of p. That is, the negation of a proposition p, denoted by ¬p, is the proposition that is false when p is true and true when p is false.
Example 1: Find out the negation of the proposition.
“Aman runs very fast.”
Solution: It is not the case that Aman runs very fast.
In simpler terms “Aman doesn't run fast”.
Example 2: Find out the negation of the proposition.
"It is raining outside."
Solution: Negation of p is- ∼p:
"It is not raining outside".
The disjunction is a connective that forms compound propositions that are false only if both statements (disjuncts) are false. It is denoted by p ∨ q, is the proposition ‘p or q’. The Disjunction p ∨ q is TRUE when anyone of p or q is TRUE.
Truth Table: Disjunction
Example 1: If p and q are two propositions where-
p: 2 + 4 = 6 and q: It is raining outside
Solution: Then, the disjunction of p and q is -p ∨ q: 2 + 4 = 6 or it is raining outside.
A conjunction is a truth-functional connective similar to "and" in English and is represented in symbolic logic with " ^ ". The conjunction of p and q, denoted by p∧q, is the proposition ‘p and q’. The Conjunction p∧q is TRUE when both p and q are TRUE.
Truth Table: Conjunction
It is a type of relationship between two statements or sentences. Denoted by ‘p → q’.
In English implication can be written as follows:
The conditional statement p → q is false when p is true and q is false, and true otherwise.
Example 1: ‘if India wins, I will give party’.
Solution: Here p is India wins.
q is I will give a party.
Truth Table: Logical Implication
In English, Bi-Conditional can be written as follows:
Truth Table: Bi-Condition
Examples:
Let ‘p’ be a compound proposition
Those compound propositions which have the same truth values in all possible cases are known as Logical Equivalences.
Denoted by ‘≡’ or ‘⇔’ symbol.
Example 1. Show that p → q and ¬p ∨ q are equivalent.
Solution. We can construct a truth table to show equivalence.
As seen from the Truth table, ‘p → q’ and ‘¬p ∨ q’ have the same truth values. So we can say p → q ≡ ¬p ∨ q.
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Example 1. Let P(x) denote the statement “x > 10″. What are the truth values of P(11) and P(5)?
Solution. P(11) means 11 > 10, which is TRUE.
P(5) means 5 > 10, which is FALSE.
Example 2. Let R(x,y) denotes the statement ” x = y + 1″. What is the truth value of the proposition R(1,3) and R(2,1) ?
Solution. R(1,3), 1 = 3 + 1 is FALSE. R(2,1), 2 = 1 + 1 is TRUE.
Example 1. Let P(x) be the statement ” x + 2 > x” . What is the truth value of the ∀xP(x).
Solution. As x+2 is greater than x for any real number, so ∀xP(x) is always TRUE.
Example 1. Let P(x) be the statement “x>5″. What is the truth value of the ∃xP(x)?
Solution. For some real number such as x = 4, ∃xP(x) is also true.
Two quantifiers are nested if one is within the scope of the other.
Example 1. ∀x∃y(x + y = 0).
Solution. In the above example, we have two quantifiers ‘∀’ and ‘∃’ within the scope of each other.
In English, we can read it as ‘for all x, there exists a y for which x+y becomes 0.’
Example 2. Translate the following statement into English.
∀x∃y(x = -y), x and y belongs to real number.
Solution. For every real number x, there is a real number y such that x = -y.
Example 3. Translate the following statement into English.
∀x∀y((x > 0) ∧ (y > 0) → (x +y > 0)) ,Domain Real Numbers.
Solution. For two positive integers, the sum of these integers is positive.
OR in simpler terms, Sum of two positive numbers is positive.
Example 1. Translate the following into logical expressions.
“If a person is a student and is computer science major, then this person takes a course in mathematics.”
Solution.
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1. What is propositional logic? |
2. What is the difference between a proposition and a predicate? |
3. What is the negation of a proposition? |
4. What is a logical implication? |
5. What is the difference between propositional and first-order logic? |
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