Question

Write the following as English sentences. Say whether they are true or false. $$ \exists a \in \mathbb{R}, \forall x \in \mathbb{R}, a x=x $$

Short answer

The English translation of the statement is: 'There exists a real number 'a' such that for all real numbers 'x', the product of 'a' and 'x' equals 'x'. This statement is true, because the real number 'a' is 1.

Step-by-step solution

Translate Mathematical Symbols into English

The symbol '∃' represents 'there exists', 'a' is a variable, '∈' symbol means 'is in' or 'belongs to', 'ℝ' stands for the set of real numbers which includes all integers, rational and irrational numbers. '∀' symbolizes 'for all', 'x' is a variable representing any number in the real number set, 'ax = x' is the condition that must be satisfied.

Construct the English Statement

Now, we can translate the mathematical expression into English: 'There exists a real number 'a' such that for all real numbers 'x', the product of 'a' and 'x' equals 'x'.

Evaluate the Statement

Ask if it is possible that there exists a real number 'a' which, when multiplied with any real number 'x', always gives the number 'x' itself back. Given the properties of multiplication, there is indeed one such real number 'a', namely 1. So, whatever real number 'x' we take, 1 * 'x' would always equal 'x'.