12 - Maths - Unit 9 - Discrete Mathematics

Explanation
The correct answer is (4) because statements are sentences that can be either true or false. In this case, options (i), (ii), and (iii) are all statements because they express facts that can be verified. However, option (iv) is not a statement because it is a classification or definition rather than a factual statement. Therefore, the correct answer is (4) because it includes all the statements.

Explanation
A compound statement made up of three simple statements will have 2^3 = 8 possible combinations of truth values for the simple statements. Therefore, the number of rows in the truth table for this compound statement will be 8.

Explanation
If p is T and q is F, then the truth value of (i) is T because it does not depend on q. The truth value of (ii) is F because it depends on q, which is F. The truth value of (iii) is T because it does not depend on q. The truth value of (iv) is T because it depends on p, which is T. Therefore, options (i), (iii), and (iv) have the truth value T.

Explanation
The given options are (1), (2), (3), and (4). The question asks for the option that is not a group. While options (1), (2), and (4) can be considered as groups since they are enclosed in parentheses, option (3) is not a group as it is not enclosed in parentheses. Therefore, the correct answer is (3).

Explanation
A monoid becomes a group if it also satisfies the inverse axiom. The inverse axiom states that for every element in the set, there exists an inverse element such that the product of the element and its inverse is equal to the identity element. This means that every element in the group has an inverse element that cancels out its effect when multiplied together. Without the inverse axiom, the monoid would not have the necessary property to be considered a group.

Explanation
In the multiplicative group of cube roots of unity, the order of an element refers to the smallest positive integer 'n' such that the element raised to the power of 'n' equals the identity element (which is 1 in this case). The cube roots of unity are 1, ω, and ω^2, where ω is a complex cube root of unity. When we raise ω to the power of 3, we get ω^3 = (ω^2)^2 = 1. Thus, the order of ω is 3. Similarly, the order of ω^2 is also 3. Therefore, the correct answer is (2) 3.

Explanation
The given question does not provide any information or context about what "the value of " is referring to. Therefore, it is impossible to determine the correct answer without additional information.

Explanation
The given answer (1) is the correct answer because it follows the order of the numbers given in the question. The numbers are listed in ascending order, starting from 1 and increasing by 3 each time. Therefore, the correct order is 1, 4, 7, 10, and so on. Among the given options, option (1) is the only one that matches this pattern.

Explanation
If every element of a group is its own inverse, then the group is abelian. This statement is correct because if every element in a group is its own inverse, it means that for every element a in the group, a * a = e, where e is the identity element. This implies that a * b = b * a for any elements a and b in the group, which is the definition of an abelian group.

Explanation
The multiplicative group of th roots of unity consists of all complex numbers that can be written as e^(2πik/n), where k is an integer and n is a positive integer. The inverse of a complex number z in this group is the complex number z^(-1), which is equal to e^(-2πik/n). Therefore, the correct answer is (3).

Explanation
The order of an element in the multiplicative group of nth roots of unity is equal to n divided by the greatest common divisor of n and the order of the element. In this case, the order of the element is 4, and the greatest common divisor of 4 and 4 is 4. Therefore, the order of the element is 4/4 = 1.