To prove this statement, we need to show that there exists a rational number that is also a power of 2. Now, let's analyze whether the statement 'some rational numbers are powers of 2' can be entailed from (1) and (2). In terms of the representations, it can be written as: I(x) ∧ P(x). Statement (2) states that 'Some integers are powers of 2.' This means that there exist integers that are also powers of 2. In terms of the representations, it can be written as: I(x) -> R(x). Statement (1) states that 'All integers are rational numbers.' This means that every integer is also a rational number. R(x) represents 'x is a rational number.'.Let's break down the given representations: Let’s finish by recapping some of the important points from this explainer.To verify whether the statement 'some rational numbers are powers of 2' can be entailed from (1) and (2), we need to analyze the given representations and statements. Hence, only option C is a rational number. Which is the quotient of two integer values where the denominator is nonzero. Thus, neither of these represents rational numbers. We have a similar story in expressions B and D. Since we cannot divide by 0, this is not a rational number. We can determine which of these expressions is rational by substituting the values of 𝑎 and 𝑏 into each expression separately.ģ 9 𝑏 𝑏 − 3 4 = 3 9 ( 3 4 ) 3 4 − 3 4. We begin by recalling that the set of rational numbers, written ℚ, is the set of all quotients of integers. Which of the following expressions is rational given 𝑎 = 1 and 𝑏 = 3 4? We can represent this information in the following Venn diagram.Įxample 6: Identifying the Rational Expression from a List of Given Expressions Represent fractions as mixed numbers for example, 5 3 = 1 2 3, which is also a rational number. We can represent numbers like this using a line It is worth noting that any decimal expansion with a finite number of digits or a repeating expansion is rational. 5, so we can also represent rational numbers as decimals. This is not the only way of representing rational numbers we have also seen that 1 2 = 0. For example, we can think about 5 3 as 5 lots of 1 3. One way of conceptualizing rational numbers like these is toĬonsider them as multiples of simpler fractions. Similarly, 5 3 and 1 2 7 are rational numbers. For example, 1 and 2 are integers, so 1 2 ∈ ℚ. We can also use this definition to find some examples of rational numbers. It is worth noting that a number cannot be rational and irrational at the same time.

We can then recall that the set of natural numbers is a subset of the integers,Īlthough it is beyond the scope of this explainer to prove this result, some numbers such as √ 2 or 𝜋 are not rationalĪnd are called irrational. This means that the set of integers isĪ subset of the set of rational numbers. First, we can note that all integers are rational numbers, Using this definition, we can see some interesting properties of the set of rational numbers. Of the form 𝑎 𝑏 where 𝑎 and 𝑏 are integers and 𝑏 is nonzero. The set of rational numbers, written ℚ, is the set of all quotients of integers.