Chapter 1: Mathematical Logic
Exercise 1.1
Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board
State which of the following sentences is a statement. Justify your answer if it is a statement. Write down its truth value.
A triangle has '\(n\)' sides
View Original SolutionIt is an open sentence. Hence, it is not a statement.
Note: Answer given in the textbook is ‘it is a statement’. However, we found that ‘It is not a statement’.
State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.
The sum of interior angles of a triangle is \(180^\circ\)
View Original SolutionIt is a statement which is true. Hence, its truth value is T.
State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.
You are amazing!
View Original SolutionIt is an exclamatory sentence. Hence, it is not a statement.
State which of the following sentences is a statement. Justify your answer if it is a statement. Write down its truth value.
Please grant me a loan.
View Original SolutionIt is a request. Hence, it is not a statement.
State which of the following sentences is a statement. Justify your answer if it is a statement. Write down its truth value.
\(\sqrt{-4}\) is an irrational number.
View Original Solution\(\sqrt{-4} = 2i\), which is a complex number, not an irrational number (irrational numbers are real). Thus, the statement is false. Hence, its truth value is F.
State which of the following sentences is a statement. Justify your answer if it is a statement. Write down its truth value.
\(x^2 - 6x + 8 = 0\) implies \(x = -4\) or \(x = -2\).
View Original SolutionThe equation \(x^2 - 6x + 8 = 0\) can be factored as \((x-2)(x-4) = 0\), so \(x = 2\) or \(x = 4\). The given implication is that \(x = -4\) or \(x = -2\), which is not true based on the roots. Therefore, it is a statement which is false. Hence, its truth value is F.
State which of the following sentences is a statement. Justify your answer if it is a statement. Write down its truth value.
He is an actor.
View Original SolutionIt is an open sentence. Hence, it is not a statement.
State which of the following sentences is a statement. Justify your answer if it is a statement. Write down its truth value.
Did you eat lunch yet?
View Original SolutionIt is an interrogative sentence. Hence, it is not a statement.
State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.
Have a cup of cappuccino.
View Original SolutionIt is an imperative sentence, hence it is not a statement.
State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.
\((x + y)^2 = x^2 + 2xy + y^2\) for all \(x, y \in \mathbb{R}\).
View Original SolutionIt is a statement which is true (algebraic identity). Hence, its truth value is T.
State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.
Every real number is a complex number.
View Original SolutionIt is a statement which is true (a real number \(a\) can be written as \(a + 0i\)). Hence, its truth value is T.
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.
View Original SolutionIt is a statement which is false (by definition, a prime number has exactly two distinct positive divisors: 1 and itself. 1 only has one positive divisor). Hence, its truth value is F.
State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.
With the sunset the day ends.
View Original SolutionIt is a statement which is true (generally understood in common parlance). Hence, its truth value is T.
State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.
\(1! = 0\)
View Original SolutionWe know \(1! = 1\). So, the statement \(1! = 0\) is false. It is a statement which is false. Hence, its truth value is F.
Note: Answer in the textbook might be incorrect if it differs.
State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.
\(3 + 5 > 11\)
View Original Solution\(3 + 5 = 8\). The statement is \(8 > 11\), which is false. It is a statement which is false. Hence, its truth value is F.
State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.
The number \(\pi\) is an irrational number.
View Original SolutionIt is a statement which is true. Hence, its truth value is T.
Note: Answer in the textbook might be incorrect if it differs.
State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.
\(x^2 - y^2 = (x + y)(x - y)\) for all \(x, y \in \mathbb{R}\).
View Original SolutionIt is a statement which is true (difference of squares identity). Hence, its truth value is T.
State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.
The number 2 is the only even prime number.
View Original SolutionIt is a statement which is true. Hence, its truth value is T.
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.
View Original SolutionIt is a statement which is true (in Euclidean geometry). Hence, its truth value is T.
State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.
The number of arrangements of 7 girls in a row for a photograph is \(7!\).
View Original SolutionIt is a statement which is true (basic permutation). Hence, its truth value is T.
State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.
Give me a compass box.
View Original SolutionIt is an imperative sentence. Hence, it is not a statement.
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.
View Original SolutionIt is an imperative sentence. Hence, it is not a statement.
State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.
It may rain today.
View Original SolutionIt is an open sentence (its truth value depends on whether it actually rains, which is uncertain at the time of utterance). Hence, it is not a statement in the context of classical logic where statements must be definitively true or false.
State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.
If \(a + b < 7\), where \(a \ge 0\) and \(b \ge 0\) then \(a < 7\) and \(b < 7\).
View Original SolutionIt is a statement which is true. If \(a \ge 0\), \(b \ge 0\) and \(a+b < 7\): Since \(b \ge 0\), then \(a < 7 - b \le 7\), so \(a < 7\). Since \(a \ge 0\), then \(b < 7 - a \le 7\), so \(b < 7\). Hence, its truth value is T.
State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.
Can you speak in English?
View Original SolutionIt is an interrogative sentence. Hence, it is not a statement.
