Saved Bookmarks
| 1. |
Show that the statement p: “If x is a real number such that x3 + 4x = 0 then x is 0” is true by1. Direct method. 2. Method of contradiction. 3.Method of contrapositive. |
|
Answer» The given compound statement is of the from “if p then q”. P: x ∈ R such that x3 + 4x = 0 q: x = 0 1. Direct method: We assume that p is true then x ∈ R such that x3 + 4x = 0 x ∈ R such that x(x2 + 4) = 0 x ∈ R such that x = 0 or x2 + 4 = 0 ⇒ x = 0 ⇒ q is true. So when p is true, q is true. Thus the given compound statement is true. 2. Method of contradiction: We assume that p is true and q is false, then x ≠ 0 x ∈ R such that x3 + 4x = 0 x ∈ R such that x(x2 + 4) = 0 x ∈ R such that x = 0 or x2 + 4 = 0 ⇒ x = 0 Which is a contradiction. So our assumption that x ≠ 0 is false. Thus the given compound statement is true. 3. Method of contrapositive. We assume that q is false, then x ≠ 0 ⇒ x ∈ R such that x3 + 4x ≠ 0 ⇒ q is false So when q is false, p is false. Thus the given compound statement is true |
|