A Proposition is a declarative sentence which is either TRUE or FALSE, but not both.
All the above are proposition as they can be true or false but not both.
Some examples of sentence which are not proposition:
All the above examples are not proposition, they can be either be true or false.
➢ Negation of Proposition
Q.1. Find out 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 runs fast”.
Truth Table: Disjunction
Truth Table: Conjunction
➢ Logical Implication
In English implication can be written as following:
The conditional statement p → q is false when p is true and q is false, and true otherwise.
Example: ‘if India wins, I will give party’.
Here p is India wins.
q is I will give party.
Truth Table: Logical Implication
In English, Bi-Conditional can be written as follows:
Truth Table: Bi-Condition
Let ‘p’ be a compound proposition
➢ Logical Equivalences
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.
➢ Some Useful Equivalence
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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.
➢ Universal Quantifier
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.
➢ Existential Quantifier
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.
➢ Negating Quantified Expression
➢ Nested quantifier
Example: ∀x∃y(x + y = 0).
In the above example we have two quantifiers ‘∀’ and ‘∃’ within scope of each other.
In English we can read it as ‘for all x, there exist a y for which x+y becomes 0.’
Example 1. 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 2. 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 number is positive.
➢ Order of Quantifier
Example 1. Translate the following into logical expression.
“If a person is a student and is computer science major, then this person takes a course in mathematics.”
► P(x) : x is student.
► Q(x) : x is computer science major.
► R(x,y) : x takes y course.
► ∀x (P(x) ∧ Q(x) ) → ∃y R(x,y).