Use of quantifiers
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This document discusses quantifiers and open statements. It defines universal and existential quantifiers and provides examples of open statements involving variables. Several quantified statements are expressed symbolically and evaluated for truth value. Universal statements about integers being perfect squares, positive, even, or divisible by 3 or 7 are determined to be true or false.
![OPEN STATEMENT
Suppose the universe consists of all integers, consider the
following open statements.
p(x): x ≤ 3 q(x): x+1 is odd r(x): x > 0
Give the truth values of the following.
i) p(2) ii) ¬ q(4) iii) p(-1) ∧ q(1)
iv) ¬p(3) ∨ r(0) v) p(0) → q(0)
vi) p(1) ↔ ¬ q(2) vii) p(4) ∨ [q(1) ∧ r(2)]
viii) p(2) ∧ [q(0) ∨ ¬r(2)]
Lakshmi R, Asst. Professor, Dept. Of ISE](https://image.slidesharecdn.com/useofquantifiers-200923092325/85/Use-of-quantifiers-3-320.jpg)
![OPEN STATEMENT
p(x): x ≤ 3 q(x): x+1 is odd r(x) > 0
i) p(2): 2 ≤ 3 ---- True
ii) ¬ q(4): 4+ 1 is not odd ---- False
iii) p(-1) ∧ q(1) : -1 ≤ 3 ∧ 1+1 is odd ----- true ∧ false = false
iv) ¬p(3) ∨ r(0) : 3 ≤ 3 ∨ 0 > 0 ---- false ∨ false = false
v) p(0) → q(0): 0 ≤ 3 → 0+1 is odd ---- true → true = true
vi) p(1) ↔ ¬ q(2): 1 ≤ 3 ↔ 2+1 is not odd ---- true ↔ false = false
vii) p(4) ∨ [q(1) ∧ r(2)]: 4 ≤ 3 ∨ [1+1 is odd ∧ 2 > 0)]
false ∨ [false ∧ true)] = false ∨ false
= false
viii) p(2) ∧ [q(0) ∨ ¬r(2)]: 2 ≤ 3 ∧ [0+1 is odd ∨ 2 ≯ 0]
true ∧ [true ∨ false] = true ∧ true
= true
Lakshmi R, Asst. Professor, Dept. Of ISE](https://image.slidesharecdn.com/useofquantifiers-200923092325/85/Use-of-quantifiers-4-320.jpg)
![p(x): x > 0
q(x): x is even
r(x): x is a perfect square
s(x): x is divisible by 3
t(x): x is divisible by 7
Lakshmi R, Asst. Professor, Dept. Of ISE
i. At least one integer is even
Sol: ∃x q(x)
ii. There exists a positive integer that is even
Sol: ∃x [p(x) ∧ q(x)]
iii. Some even integers are divisible by 3
Sol: ∃x [q(x) ∧ s(x)]
iv. Every integer is either even or odd
Sol: ∀x [q(x) ⊻ ¬q(x)]
v. If x is even and a perfect square, then x is not divisible by 3
Sol: ∀x [q(x) ∧ r(x)] → ¬s(x)
vi. If x is odd or is not divisible by 7, then x is divisible by 3
Sol: ∀x [¬ q(x) ∨ ¬ t(x)] → s(x)](https://image.slidesharecdn.com/useofquantifiers-200923092325/85/Use-of-quantifiers-7-320.jpg)
![2. Consider the open statements
given below.
p(x): x > 0
q(x): x is even
r(x): x is a perfect square
s(x): x is divisible by 3
t(x): x is divisible by 7
Lakshmi R, Asst. Professor, Dept. Of ISE
Express each of the following
symbolic statements in words
and indicate its truth value
i. ∀x [r(x) → p(x)]
ii. ∃x [s(x) ∧ ¬ q(x)]
iii. ∀x [¬ r(x)]
iv. ∀x [r(x) ∨ t(x)]](https://image.slidesharecdn.com/useofquantifiers-200923092325/85/Use-of-quantifiers-8-320.jpg)
![p(x): x > 0
q(x): x is even
r(x): x is a perfect square
s(x): x is divisible by 3
t(x): x is divisible by 7
Lakshmi R, Asst. Professor, Dept. Of ISE
i. ∀x [r(x) → p(x)]
Sol: “ for all integers x, if x is a perfect squares, then x > 0”
Let x = 0.
r(0): 0 is a perfect square.This statement is true. 02 = 0
p(0): 0 > 0. this statement is false.
∴ ∀x [r(x) → p(x)] = ∀x [true → false] is a false statement
ii. ∃x [s(x) ∧ ¬ q(x)]
Sol: “there exists some integers x such that x is divisible by 3
and x is not even”
Let x = 9
S(9): 9 is divisible by 3.This statement is true.
¬ q(x): 9 is not even.This statement is true
∴ ∃x [s(x) ∧ ¬ q(x)]= ∃x [true ∧ true] is a true statement.](https://image.slidesharecdn.com/useofquantifiers-200923092325/85/Use-of-quantifiers-9-320.jpg)
![p(x): x > 0
q(x): x is even
r(x): x is a perfect square
s(x): x is divisible by 3
t(x): x is divisible by 7
Lakshmi R, Asst. Professor, Dept. Of ISE
iii. ∀x [¬ r(x)]
Sol: “for any integer x,x is not a perfect square”
Let x = 9.
¬ r(9): 9 is not a perfect square.This is false.
∴ ∀x [¬ r(x)] is a false statement
iv. ∀x [r(x) ∨ t(x)]
Sol: “for any integer x, x is a perfect square or x is divisible
by 7”
Let x = 2.
r(2): 2 is perfect square.This is false
T(2): 2 is divisible by 7.This is false
∴ ∀x [r(x) ∨ t(x)] = ∀x [false ∨ false] is a false statement](https://image.slidesharecdn.com/useofquantifiers-200923092325/85/Use-of-quantifiers-10-320.jpg)
Use of quantifiers
- 1. USE OF QUANTIFIERS By, Lakshmi R Asst. professor, Dept. of ISE
- 2. OPEN STATEMENT It is a declarative statement which contains one or more variable. It is not a statement, but when the variables in it are replaced by certain allowable choices, it can be called as a statement. Eg: i) x + 5 = 10 ii) x3 < 100 Open statements containing a variable are denoted by p(x), q(x), etc.. x is called a free variable. Eg: p(x): x3 < 100. if x = 4, p(x): 64 < 100 Truth value of p(4) is true Lakshmi R, Asst. Professor, Dept. Of ISE
- 3. OPEN STATEMENT Suppose the universe consists of all integers, consider the following open statements. p(x): x ≤ 3 q(x): x+1 is odd r(x): x > 0 Give the truth values of the following. i) p(2) ii) ¬ q(4) iii) p(-1) ∧ q(1) iv) ¬p(3) ∨ r(0) v) p(0) → q(0) vi) p(1) ↔ ¬ q(2) vii) p(4) ∨ [q(1) ∧ r(2)] viii) p(2) ∧ [q(0) ∨ ¬r(2)] Lakshmi R, Asst. Professor, Dept. Of ISE
- 4. OPEN STATEMENT p(x): x ≤ 3 q(x): x+1 is odd r(x) > 0 i) p(2): 2 ≤ 3 ---- True ii) ¬ q(4): 4+ 1 is not odd ---- False iii) p(-1) ∧ q(1) : -1 ≤ 3 ∧ 1+1 is odd ----- true ∧ false = false iv) ¬p(3) ∨ r(0) : 3 ≤ 3 ∨ 0 > 0 ---- false ∨ false = false v) p(0) → q(0): 0 ≤ 3 → 0+1 is odd ---- true → true = true vi) p(1) ↔ ¬ q(2): 1 ≤ 3 ↔ 2+1 is not odd ---- true ↔ false = false vii) p(4) ∨ [q(1) ∧ r(2)]: 4 ≤ 3 ∨ [1+1 is odd ∧ 2 > 0)] false ∨ [false ∧ true)] = false ∨ false = false viii) p(2) ∧ [q(0) ∨ ¬r(2)]: 2 ≤ 3 ∧ [0+1 is odd ∨ 2 ≯ 0] true ∧ [true ∨ false] = true ∧ true = true Lakshmi R, Asst. Professor, Dept. Of ISE
- 5. QUANTIFIERS Statements that contain phrases that are associated with the idea of quantity are called quantifiers. Types of quantifiers: Universal Quantifiers: denoted by ∀x – “for all x”, “for any x”, “for each x”, “for every x” Eg: ∀x ∈ s, p(x): for all x belong to S. Where, p(x) is an open statement Existential quantifiers: denoted by ∃x – “for some x”, “for at least one x”, “there exists an x” Lakshmi R, Asst. Professor, Dept. Of ISE
- 6. 1. For the universe of all integers, let p(x): x > 0 q(x): x is even r(x): x is a perfect square s(x): x is divisible by 3 t(x): x is divisible by 7 Lakshmi R, Asst. Professor, Dept. Of ISE Write the following quantified statements in symbolic form. i. At least one integer is even ii. There exists a positive integer that is even iii. Some even integers are divisible by 3 iv. Every integer is either even or odd v. If x is even and a perfect square, then x is not divisible by 3 vi. If x is odd or is no divisible by 7, then x is divisible by 3
- 7. p(x): x > 0 q(x): x is even r(x): x is a perfect square s(x): x is divisible by 3 t(x): x is divisible by 7 Lakshmi R, Asst. Professor, Dept. Of ISE i. At least one integer is even Sol: ∃x q(x) ii. There exists a positive integer that is even Sol: ∃x [p(x) ∧ q(x)] iii. Some even integers are divisible by 3 Sol: ∃x [q(x) ∧ s(x)] iv. Every integer is either even or odd Sol: ∀x [q(x) ⊻ ¬q(x)] v. If x is even and a perfect square, then x is not divisible by 3 Sol: ∀x [q(x) ∧ r(x)] → ¬s(x) vi. If x is odd or is not divisible by 7, then x is divisible by 3 Sol: ∀x [¬ q(x) ∨ ¬ t(x)] → s(x)
- 8. 2. Consider the open statements given below. p(x): x > 0 q(x): x is even r(x): x is a perfect square s(x): x is divisible by 3 t(x): x is divisible by 7 Lakshmi R, Asst. Professor, Dept. Of ISE Express each of the following symbolic statements in words and indicate its truth value i. ∀x [r(x) → p(x)] ii. ∃x [s(x) ∧ ¬ q(x)] iii. ∀x [¬ r(x)] iv. ∀x [r(x) ∨ t(x)]
- 9. p(x): x > 0 q(x): x is even r(x): x is a perfect square s(x): x is divisible by 3 t(x): x is divisible by 7 Lakshmi R, Asst. Professor, Dept. Of ISE i. ∀x [r(x) → p(x)] Sol: “ for all integers x, if x is a perfect squares, then x > 0” Let x = 0. r(0): 0 is a perfect square.This statement is true. 02 = 0 p(0): 0 > 0. this statement is false. ∴ ∀x [r(x) → p(x)] = ∀x [true → false] is a false statement ii. ∃x [s(x) ∧ ¬ q(x)] Sol: “there exists some integers x such that x is divisible by 3 and x is not even” Let x = 9 S(9): 9 is divisible by 3.This statement is true. ¬ q(x): 9 is not even.This statement is true ∴ ∃x [s(x) ∧ ¬ q(x)]= ∃x [true ∧ true] is a true statement.
- 10. p(x): x > 0 q(x): x is even r(x): x is a perfect square s(x): x is divisible by 3 t(x): x is divisible by 7 Lakshmi R, Asst. Professor, Dept. Of ISE iii. ∀x [¬ r(x)] Sol: “for any integer x,x is not a perfect square” Let x = 9. ¬ r(9): 9 is not a perfect square.This is false. ∴ ∀x [¬ r(x)] is a false statement iv. ∀x [r(x) ∨ t(x)] Sol: “for any integer x, x is a perfect square or x is divisible by 7” Let x = 2. r(2): 2 is perfect square.This is false T(2): 2 is divisible by 7.This is false ∴ ∀x [r(x) ∨ t(x)] = ∀x [false ∨ false] is a false statement