Question

Using elementary mathematics, determine the truth values of the following conjunctions: (1) Nine is an odd number less than four. (2) Eight is greater than one but less than fifteen. (3) A rhombus has four equal sides, two of which are parallel. (4) Seven and ten are prime numbers.

Short answer

Conjunction 1 is false, Conjunction 2 is true, Conjunction 3 is false, and Conjunction 4 is false.

Step-by-step solution

Conjunction 1

The conjunction is: "Nine is an odd number less than four." This conjunction has two parts: (1) Nine is an odd number. (2) Nine is less than four. By definition, an odd number is an integer that is not divisible by 2. Nine does fit this description, so the first part is true. However, nine is not less than four. Thus, the second part is false. Since not all parts of the conjunction are true, the conjunction is false.

Conjunction 2

The conjunction is: "Eight is greater than one but less than fifteen." This conjunction has two parts: (1) Eight is greater than one. (2) Eight is less than fifteen. Eight is greater than one, so the first part is true. Eight is also less than fifteen, so the second part is true. Both parts of the conjunction are true, so the conjunction is true.

Conjunction 3

The conjunction is: "A rhombus has four equal sides, two of which are parallel." This conjunction has two parts: (1) A rhombus has four equal sides. (2) A rhombus has two parallel sides. A rhombus does indeed have four equal sides, so the first part is true. However, the second part is false because a rhombus has all of its sides parallel, not just two. Therefore, the conjunction is false.

Conjunction 4

The conjunction is: "Seven and ten are prime numbers." This conjunction has two parts: (1) Seven is a prime number. (2) Ten is a prime number. A prime number is an integer greater than one that has exactly two distinct positive divisors: 1 and itself. Seven does fit this definition, so the first part is true. However, ten is not a prime number because it has four positive divisors: 1, 2, 5, and 10. Thus, the second part is false. Since not all parts of the conjunction are true, the conjunction is false.

Key concepts

Understanding Elementary Mathematics

Elementary mathematics encompasses the basic concepts of arithmetic, including whole numbers, odd and even numbers, basic operations, and elementary number theory, such as understanding prime numbers. When we talk about odd numbers, these are integers not divisible evenly by 2, like 1, 3, 5, and so forth.

Prime numbers, on the other hand, are those greater than 1 that cannot be formed by multiplying two smaller natural numbers. Seven, for instance, is prime because the only numbers that can multiply to give 7 are 1 and 7 itself. The analysis of truth values in our exercise involves evaluating statements to see if they are true or false based on these elementary concepts.

Logical Conjunctions in Mathematics

Logical conjunctions are statements joined by the word 'and' and are only true if both statements are true. In the context of mathematics, conjunctions are used to combine multiple mathematical statements. For example, 'Eight is greater than one but less than fifteen' joins two statements with 'but' which functions as 'and' does in logic.

The conjunction is considered true if both 'Eight is greater than one' and 'Eight is less than fifteen' are true. The use of conjunctions in mathematical statements allows complex conditions to be succinctly expressed in one sentence.

Properties of Numbers

In the realm of numbers, several properties define and categorize them. For example, in our exercise, we discuss odd numbers, which have the property that they cannot be divided evenly by two, and prime numbers, which are defined by having only two divisors: one and itself.

We also come across composite numbers, such as ten in the exercise, which have more than two divisors. Understanding properties like these is vital for determining the truth values of mathematical statements. Mistaking these properties can result in incorrect evaluations, as seen with the misunderstanding about ten being a prime number.

Geometry Concepts: The Rhombus

Geometric shapes are foundational in geometry, each with unique properties. A rhombus, mentioned in our exercise, is a four-sided shape with all sides of equal length, which is correctly recognized in the exercise's solution. However, an essential aspect of the rhombus is that all opposite sides are parallel, not just two.

This incorrect statement in the exercise represents a common misunderstanding of geometric properties. It is important to recognize that geometry is not just about shapes, but also about understanding the very specific characteristics that define those shapes. Misinterpretations of such properties can alter the validity of logical statements involving geometry.