Using truth table prove that p ˅ (q ˄ r) ≡ (p ˅ q) ˄ (p ˅ r). - Mathematics and Statistics

Advertisements
Advertisements
Chart
Sum

Using truth table prove that p ˅ (q ˄ r) ≡ (p ˅ q) ˄ (p ˅ r).

Advertisements

Solution

p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

1 2 3 4 5 6 7 8
p q r q ∧ r p ∨ (q ∧ r) p ∨ q p ∨ r (p ∨ q) ∧ (p ∨ r)
T T T T T T T T
T T F F T T T T
T F T F T T T T
T F F F T T T T
F T T T T T T T
F T F F F T F F
F F T F F F T F
F F F F F F F F

The entries in the columns 5 and 8 are identical.

∴ p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

  Is there an error in this question or solution?
Chapter 1.1: Mathematical Logic - Q.5

RELATED QUESTIONS

If p ˄ q = F, p → q = F, then the truth value of p and q is ______.


Write truth values of the following statements: ∃ n ∈ N such that n + 5 > 10.


Write the following statement in symbolic form and find its truth value:
∀ n ∈ N, n2 + n is an even number and n2 - n is an odd number.


Using truth tables, examine whether the statement pattern (p ∧ q) ∨ (p ∧ r) is a tautology, contradiction or contingency.


If p, q, r are the statements with truth values T, F, T, respectively then find the truth value of (r ∧ q) ↔ ∼ p


State which of the following is the statement. Justify. In case of a statement, state its truth value.

x – 3 = 14


State which of the following is the statement. Justify. In case of a statement, state its truth value.

Zero is a complex number.


State which of the following is the statement. Justify. In case of a statement, state its truth value.

Please get me breakfast.


State which of the following is the statement. Justify. In case of a statement, state its truth value.

Congruent triangles are similar.


State which of the following is the statement. Justify. In case of a statement, state its truth value.

Do you like Mathematics?


State which of the following is the statement. Justify. In case of a statement, state its truth value.

The sunsets in the west


State which of the following is the statement. Justify. In case of a statement, state its truth value.

x2 – 6x – 7 = 0, when x = 7


State which of the following is the statement. Justify. In case of a statement, state its truth value.

The sum of cube roots of unity is zero.


Write the truth values of the following.

4 is odd or 1 is prime.


Write the truth values of the following.

64 is a perfect square and 46 is a prime number.


Write the truth values of the following.

5 is a prime number and 7 divides 94.


Write the truth values of the following.

It is not true that 5−3i is a real number.


Write the truth value of the following.

If 3 × 5 = 8 then 3 + 5 = 15.


Write the truth value of the following.

Milk is white if and only if sky is blue.


If the statement p, q are true statement and r, s are false statement then determine the truth value of the following:

(p → q) ∨ (r → s)


If the statement p, q are true statement and r, s are false statement then determine the truth value of the following:

(q ∧ r) ∨ (∼ p ∧ s)


If the statement p, q are true statement and r, s are false statement then determine the truth value of the following:

[∼ p ∧ (∼ q ∧ r)] ∨ [(q ∧ r) ∨ (p ∧ r)]


If the statement p, q are true statement and r, s are false statement then determine the truth value of the following:

[(∼ p ∧ q) ∧ ∼ r] ∨ [(q → p) → (∼ s ∨ r)]


If A = {3, 5, 7, 9, 11, 12}, determine the truth value of the following.

∀ x ∈ A, x2 + x is an even number


If p ∧ q is F, p → q is F then the truth values of p and q are ________.


Select and write the correct answer from the given alternative of the following question:

If A = {1, 2, 3, 4, 5} then which of the following is not true?


Write the truth value of the following statement:

`sqrt5` is an irrational but `3sqrt5` is a complex number.


Write the truth value of the following statement:

∀ n ∈ N, n2 + n is even number while n2 – n is an odd number.


Write the truth value of the following statement:

∃ n ∈ N such that n + 5 > 10.


State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

Did you eat lunch yet?


State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

1 is a prime number.


State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

3 + 5 > 11


State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

Two co-planar lines are either parallel or intersecting.


State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

Bring the motor car here.


Choose the correct alternative :

Which of the following is not a statement?


Which of the following statements is false?


Choose the correct alternative :

For the following three statements
p : 2 is an even number.
q : 2 is a prime number.
r : Sum of two prime numbers is always even.
Then, the symbolic statement (p ∧ q) → ∼ r means.


Choose the correct alternative :

If p : He is intelligent.
q : He is strong
Then, symbolic form of statement “It is wrong that, he is intelligent or strong” is


The negation of the proposition “If 2 is prime, then 3 is odd”, is ______ 


Choose the correct alternative :

The statement (∼ p ∧ q) ∨∼ q is


Choose the correct alternative:

Which of the following is always true?


Choose the correct alternative :

∼ (p ∨ q) ∨ (∼ p ∧ q) is logically equivalent to


Choose the correct alternative :

If p is the sentence ‘This statement is false’ then


Choose the correct alternative :

Conditional p → q is equivalent to


Fill in the blanks :

Truth value of if x = 2, then x2 = − 4 is –––––––––.


Fill in the blanks :

p ↔ q is false when p and q have ––––––––– truth values.


Fill in the blanks :

Let p : the problem is easy. r : It is not challenging then verbal form of ∼ p → r is –––––––––.


State whether the following statement is True or False :

“His birthday is on 29th February” is not a statement.


State whether the following statement is True or False :

x2 = 25 is true statement.


State whether the following statement is True or False :

Truth value of `sqrt(5)` is not an irrational number is T.


State whether the following statement is True or False :

p ∧ t = p.


Solve the following :

State which of the following sentences are statements in logic.
Ice cream Sundaes are my favourite.


Solve the following :

State which of the following sentences are statements in logic.
x + 3 = 8 ; x is variable.


Solve the following :

State which of the following sentences are statements in logic.
z is a positive number.


Solve the following :

State which of the following sentences are statements in logic.
(2 + 1)2 = 9.


Solve the following :

State which of the following sentences are statements in logic.
Why are you sad?


Solve the following :

State which of the following sentences are statements in logic.
The square of any odd number is even.


Solve the following :

State which of the following sentences are statements in logic.
All integers are natural numbers.


Solve the following :

State which of the following sentences are statements in logic.
If x is real number then x2 ≥ 0.


Solve the following :

State which of the following sentences are statements in logic.
What a horrible sight it was!


Which of the following sentence is a statement? In case of a statement, write down the truth value.

The square of every real number is positive.


Which of the following sentence is a statement? In case of a statement, write down the truth value.

Every parallelogram is a rhombus.


Which of the following sentence is a statement? In case of a statement, write down the truth value.

a2 − b2 = (a + b) (a − b) for all a, b ∈ R.


Which of the following sentence is a statement? In case of a statement, write down the truth value.

Please carry out my instruction.


Which of the following sentence is a statement? In case of a statement, write down the truth value.

The Himalayas is the highest mountain range.


Which of the following sentence is a statement? In case of a statement, write down the truth value.

(x − 2) (x − 3) = x2 − 5x + 6 for all x∈R.


Which of the following sentence is a statement? In case of a statement, write down the truth value.

What are the causes of rural unemployment?


Assuming the following statement.
p : Stock prices are high.
q : Stocks are rising.
to be true, find the truth value of the following.

Stock prices are not high or stocks are rising.


If A = {2, 3, 4, 5, 6, 7, 8}, determine the truth value of the following statement.

∃ x ∈ A, such that 3x + 2 > 9


If p is any statement then ( p ˅ ∼ p) is a


State the truth value of `sqrt(3)` is not an irrational number


State the truth value of (p ˅ ∼p)


State the truth value of (p ˄ ∼p) 


If statements p, q are true and r, s are false, determine the truth values of the following.

~ p ∧ (q ∨ ~ r)


If statements p, q are true and r, s are false, determine the truth values of the following.

(p ∧ ~r) ∧ (~q ∨ s)


State whether the following statement is True or False:

p → q is equivalent to p → ~ q


State whether the following statement is True or False:

(p ˅ q) ˄ ~ p is a contradiction


State whether the following statement is True or False:

p ↔ q is false when p and q have different truth values


State whether the following statement is True or False:

p ˅ ~ p ≡ ~ c


The truth value of the statement “Neither 27 is a prime number nor divisible by 4” is ______


Using truth table prove that ~ p ˄ q ≡ ( p ˅ q) ˄ ~ p


Without using truth table show that

(p ∨ q) ∧ (~ p ∨ ~ q) ≡ (p ∧ ~ q) ∨ ( ~ p ∧ q)


Which of the following quantified statement is true?


Let a: ~ (p ∧ ~ r) v (~ q v s) and

b: (p v s) ↔ (q ∧ r).

If the truth values of p and q are true and that of rands are false, then the truth values of a and bare respectively.


The negation of 'For every real number x, `x^2 ≥ 0`' is ______ 


If (p ∧ ~ q) → ~ p is false, the truth values of p and q are respectively.


If the truth value of statement (q ∧ ~ r) → p is false (F), then the truth values of the statements p, q, rare respectively.


If p ↔ (~ p → q) is false, then the truth values of p and q are respectively ______.


If q Λ ∼p is T, then which of the following is correct?


If truth values of p ↔ r, r, p ↔ q are F, T, F respectively, then respective truth values of p and q are ______.


If p `rightarrow` (p ∧ ∼q) is false, then the truth values of p and q are respectively ______.


Consider the following two statements.

Statement p:

The value of sin 120° can be divided by taking θ = 240° in the equation 2 sin `θ/2` = `sqrt(1 + sin θ) - sqrt(1 - sinθ)`.

Statement q:

The angles A, B, C and D of any quadrilateral ABCD satisfy the equation `cos(1/2(A + C)) + cos(1/2(B + D))` = 0

Then the truth values of p and q are respectively.


If p : Every square is a rectangle. q : Every rhombus is a kite, then truth values of p `rightarrow` q and p `leftrightarrow` q are ______ and ______ respectively.


The position vector of points A and B are `6bar(a) + 2bar(b) and bar(a) -3bar(b)` . If the point C divides AB in the ratio 3 : 2 then show that the position vector of C is 3`bar(a)- bar(b)`.


Share
Notifications



      Forgot password?
Use app×