show that the following statements are not true. p : If all the angles of a triangle are equal, then the triangle is an obtuse angled triangle. q : The equation x – = does not have a root lying between and . MATHEMATICAL REASONING .
Which of the following statements are true and which are false? In each case give a valid reason for saying so. p: Each radius of a circle is a chord of the circle. q: The centre of a circle bisects each chord of the circle.
(iii) r: Circle is a particular case of an ellipse. (iv) s: If x and y are integers such that x > y , then – x < – y . (v) t : is a rational number. Miscellaneous Examples Example Check whether “Or” used in the following compound statement is exclusive or inclusive?
Write the component statements of the compound statements and use them to check whether the compound statement is true or not. Justify your answer. t: you are wet when it rains or you are in a river. Solution “Or” used in the given statement is inclusive because it is possible that it rains and you are in the river.
The component statements of the given statement are p : you are wet when it rains. q : You are wet when you are in a river. Here both the component statements are true and therefore, the compound statement is true. Example Write the negation of the following statements: p: For every real number x , x > x.
q: There exists a rational number x such that x = . (iii) r: All birds have wings. (iv) s: All students study mathematics at the elementary level. Solution (i) The negation of p is “It is false that p is” which means that the condition x > x does not hold for all real numbers.
This can be expressed as ∼ p: There exists a real number x such that x < x. Negation