Mathematical Statements, Proofs, and Implications
A mathematical statement is a meaningful sentence that has a definite state of being true or false.
Statement? Truth Value
7 > 3 Yes True
2 = π Yes False
Is 5 an even No; this is a question. N/A
χ – 3 = 2 No; this is an open N/A
Let χ be a real Yes; χ is defined. True
number. If χ > 0,
then χ > 0
* An open sentence is a sentence which can be a statement if its variables are defined with appropriate values.
A proof establishes the truth of a given statement beyond doubt.
Statement: For every real number χ, χ + 1 ≥ 2χ.
Let χ be a real number.
Thus, (χ – 1) is also a real number.
Hence, (χ – 1) ≥ 0.
Then, χ - 2χ +1 ≥ 0.
So, χ + 1