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Multiple choice mathematical modelling proof by contradiction similar triangles

The proposition $(p \, \Rightarrow \, ~p)\,\wedge \, (~p \, \Rightarrow  \, p)$ is a

  1. Tautology

  2. neither tautology nor contradiction

  3. contradiction

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
p ~p $P\Rightarrow   ~P$ $~P\Rightarrow P$ $(P\Rightarrow ~P)\wedge (~P\Rightarrow P)$
T F  F   T                    F
F T  T F F
Multiple choice mathematical modelling proof by contradiction similar triangles

Which of the following is true for counter example.

  1. A counter example is an exception to a proposed general rule or law

  2. A counter example is a specific instance of the falsity of a universal quantification (a "for all" statement).

  3. Any hard-working student is a counter example to "all students are lazy"

  4. None of these
Reveal answer Fill a bubble to check yourself
A,B,C Correct answer
Explanation

All three a,b,c are true

A counter example is an exception to a proposed general rule or law is a defination.Counterexamples are often used in math to prove the boundaries of possible theorems.option C is an example of counterexample and option b and option a are more or less same