A divides b relationship

Example 5 - R = {(a, b) : 2 divides a-b} is equivalence relation

Given: Two odd numbers a and b. Prove: a+b is even. From definition of odd number, a=2n+1 and b=2k+1 for integers n, k. a+b = 2n+1+2k+1 = 2n + 2k+ 2. Figure given below shows a relationship between the steps P and Q. Write this relation (i) in Let R be the relation on A defined by {(a, b): a ∈ A, a divides b}. 5 Set Theory Exercise 1; 6 Relationships between Sets .. On the left, the sets A and B are disjoint, because the loops don't overlap. . like 'add', 'subtract', ' multiply' and 'divide', so we can combine two sets to form a third set in various ways.

Points inside the rectangle represent elements that are in the universal set; points outside represent things not in the universal set. You can think of this rectangle, then, as a 'fence' keeping unwanted things out - and concentrating our attention on the things we're talking about.

Other sets are represented by loops, usually oval or circular in shape, drawn inside the rectangle. Again, points inside a given loop represent elements in the set it represents; points outside represent things not in the set.

On the right A is a subset of B, because the loop representing set A is entirely enclosed by loop B. Worked Examples[ edit ] Venn diagrams: Note that the rectangle representing the universal set is divided into four regions, labelled i, ii, iii and iv. What can be said about the sets A and B if it turns out that: So A is a subset of B, and the diagram should be re-drawn like Fig 2 above.

definition - What does it mean to say "a divides b" - Mathematics Stack Exchange

The diagram should then be re-drawn like Fig 1 above. Example 2 a Draw a Venn diagram to represent three sets A, B and C, in the general case where nothing is known about possible relationships between the sets. In each case, the Venn diagram can be re-drawn so that empty regions are no longer included.

If region ii is empty, the loop representing A should be made smaller, and moved inside B and C to eliminate region ii. If regions ii, iii and iv are empty, make A and B smaller, and move them so that they are both inside C thus eliminating all three of these regionsbut do so in such a way that they still overlap each other thus retaining region vi.

If regions iii and vi are empty, 'pull apart' loops A and B to eliminate these regions, but keep each loop overlapping loop C. Drawing Venn diagrams for each of the above examples is left as an exercise for the reader. Example 3 The following sets are defined: The technique is as follows: Draw a 'general' 3-set Venn diagram, like the one in Example 2.

Go through the elements of the universal set one at a time, once only, entering each one into the appropriate region of the diagram.

Discrete Mathematics/Set theory

Re-draw the diagram, if necessary, moving loops inside one another or apart to eliminate any empty regions. Don't begin by entering the elements of set A, then set B, then C — you'll risk missing elements out or including them twice!

The key was that we had played together for a long time. There were a lot of injuries in the buildup — David Beckham was coming back from a metatarsal, Kieron Dyer and Steven Gerrard were both affected by injury and replacements were flying in and out. You want to get rid of all that razzmatazz — a player needs to have played at least a couple of games at the end of the season, and the manager needs to be strong enough to make that call.

• ‘On the pitch there was a divide’: tales behind England’s World Cup failures
• Example 5 - Chapter 1 Class 12 Relation and Functions
• Equivalence Relation on Set

I have good memories of that World Cup. I think it was the most well-organised tournament I went to. There were some big distances to travel but the facilities were fantastic, the pitches were first class and I think every detail was taken care of perfectly.

My family came out, and nobody had a murmur of complaint. Plus we beat Argentina.

Jermaine Jenas Knocked out by Portugal on penalties in quarter-finals We were staying in Baden-Baden and a lot of us had friends and relatives nearby. It felt good at the time. But the way the British media reported the fact that our families were out there and enjoying themselves, the way they blew everything out of proportion, I thought was out of order.

My mum was having conversations with journalists, having meals with them — they were staying at the same hotel — but then these reporters were stabbing people in the back and sneaking secret cameras around. Since then the FA have gone down a different route.