Please see the attached file for the fully formatted problems.
1: Use the Euclidean Algorithm to find the Highest Common Factor of 1176 and
2: Use the rules of natural deduction to prove ((A --> (B -->C)) -->((A B)--> C)).
3: Use the rules of natural deduction to prove (A (B U C)) ((A B) C)
4:Determine which of the following functions are one-one and which are onto
i) f:R?>R f(s) =3x+1
ii) g: N?>N, f(s) =4x+1
iii) h:R?>R. h(s) =x2-x
5: Given two functions f B ?> C and g.A ?> B such that f°g:A ?> C is onto prove that f is onto.
Give an example in which fog is onto but g is not.
6: Determine which of the following relations,[xpyl on the set N of natural numbers are (i) Reflexive, (ii) symmetric (iii transitive.(iv) antisymmetric, justifying your answers.
(a) x is an integer multiple of y
7: Define a relation p on the set Z by xpy iff x-y is a multiple of 5.
Prove p that is an equivalence relation and describe the equivalence classes of p.
1. 2 | 1176 1960
2 |588 980
Since there are no factors other than 1 between 299 and 490,
the highest common factor of 1176 and 1960 is 2*2=4.
2. Since A=>(B=>C) and B=>C,
3. Let us denote the union of A and B by AUB. Then A(BUC) means all elements belong to A but not to BUC, Of course not belong to either B or C. So every element in A(BUC) belongs to A but not to B meaning belonging to AB, at the same time it does not belong to C, so every element in A(BUC) belongs to (AB)C.
4.(i) This function is one-to-one and onto. Since, let y1=f(x1) and y2=f(x2), if y1=y2, then we have 3x1+1=3x2+1, so x1=x2, which means this function f(x)=3x+1 is ...
A selection of foundation of math problems are solved.