 Hello and welcome to the session. In this session first we will discuss implications. Here we shall discuss the implications of if then only if and if and only if. If the statement is of the form if p then q then this can be written in the following ways like p implies q or p implies q. This symbol is for implies. The next form in which if p then q statement can be written is p is a sufficient condition for q. Next is p only if q then we have q is a necessary condition for p then we have negation q implies negation p. Consider the statement if x is a prime number then x is odd. This is of the form if p then q. So here the statement p is x is a prime number and the statement q is x is odd. So this statement can be written in five forms. According to the first form we have p implies q that is x is a prime number implies x is odd then for x to be odd it is sufficient that x is a prime number then x is a prime number only if x is odd then for x to be a prime number x is odd is a necessary condition x is not odd implies that x is not a prime number. Next we discuss contrapositive of a statement. If a statement is of the form if p then q then its contrapositive is given by if negation q then negation p. Like consider this statement again if x is a prime number then x is odd here this is the statement p this is the statement q. Now the contrapositive of this statement is given by if negation q that is if x is not odd then negation p that is then x is not a prime number. This is the contrapositive of the given statement. Next we have the converse of a statement again if the statement is of the form if p then q then its converse is given by if q then p. Consider this statement if x is a prime number then x is odd converse of the statement is given by if q that is if x is odd then p that is then x is a prime number this is the converse of the given statement. Next we shall discuss if and only if this is represented by this symbol it means the following equivalent forms for the given statements p and q like the first one says p if and only if q then next is q if and only if p next one is p is necessary and sufficient condition for q and vice versa then this can also be written as p if and only if q. Consider a statement p given by if it rains then I stay at home and a statement q given by if I stay at home then it rains then p if and only if q is given by it rains if and only if I stay at home. Next we shall discuss validating statements here we shall discuss some techniques to find when a statement is valid like if the statement is of the form p and q then to show that p and q is true we will perform the following steps according to the first step we have we show that the statement p is true and in the second step we show that the statement q is true. So if we show that both the statements p and q are true then the statement p and q would be true. Next if the statement is of the form p or q then to show that p or q is true we must consider the following like the case one which says that by assuming that p is false we show that q must be true then by assuming q is false we show that p must be true. The next we have if the statement is of the form if p then q then to prove this statement we need to show any one of the following cases the first one says by assuming that p is true we show that q must be true or the second case that we can show is that by assuming q is false we show that p must be false. This first case is the direct method and the second case is the contra positive method then if the statement is of the form p if and only if q then to prove this statement we need to show if p is true then q is true if q is true then p is true. Then next we have method of contradiction if we have given a statement p then to check whether the statement p is true what we do is we assume that p is not true that is we say that negation p is true then we arrive at some result which contradicts our assumption and thus we can conclude that p is true. Next method to check the validity of statement is using a counter example by this method we can show that a statement is false this method involves giving example of a situation where the statement is not valid and such an example is called the counter example the name itself suggests that this is an example to counter the given statement let's consider a statement if x is an integer x square is even then x is also even this is of the form if p then q here the statement p is given by x is an integer and x square is even statement q is given by x is even we will try to prove the given statement to be true using the counter positive method in which we assume that q is false that is we take that negation q is true that is we say that x is odd then we need to show that negation q implies negation p this is what we need to show we have assumed that negation q is true now we will show that negation p is true since we have taken that x is odd so we can write x as 2n plus 1 for some integer n then x square would be equal to 2n plus 1 the whole square which is equal to 4n square plus 4n plus 1 now this number is odd due to the presence of this one so we say that when x is odd we get x square is also odd since we are getting that x square is odd so we can say that negation p is true hence we get that negation q implies negation p thus the given statement is true so this is how we check the validity of a statement using different methods this completes the session hope you understood the implications and validating statements